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1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Time s" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 26 "An introduction to series " }} {PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }} {PARA 0 "" 0 "" {TEXT -1 18 "Version: 30.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 22 "A sum of odd numbers " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 8 "Ques tion" }{TEXT -1 55 ": Find the sum of the first 100 positive odd integ ers. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 8 " Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 67 "It is not ne cessary to add all the numbers in the conventional way." }}{PARA 0 "" 0 "" {TEXT -1 15 "Suppose we let " }{TEXT 268 1 "S" }{TEXT -1 46 " sta nd for the sum that we are trying to find." }}{PARA 0 "" 0 "" {TEXT -1 72 "The sequence of odd numbers starting with the number 1 has gene ral term " }{XPPEDIT 18 0 "a[n]=2*n-1" "6#/&%\"aG6#%\"nG,&*&\"\"#\"\" \"F'F+F+F+!\"\"" }{TEXT -1 40 ", so the 100 th positive odd integer is " }{XPPEDIT 18 0 "a[100]=199" "6#/&%\"aG6#\"$+\"\"$*>" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 263 1 "S" }{TEXT -1 33 " = 1 + 3 + 5 + 7 + \+ " }{TEXT 264 5 ". . ." }{TEXT -1 27 " + 193 + 195 + 197 + 199. " }} {PARA 0 "" 0 "" {TEXT -1 18 "We may also write " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 265 1 "S" }{TEXT -1 27 " = 199 + 197 + 195 + 193 + " }{TEXT 266 5 ". . ." }{TEXT -1 36 " + 7 + 5 + 3 + \+ 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Now consider adding these two sums together to form " }{XPPEDIT 18 0 "2*S" "6#*&\"\"#\"\"\"%\"SGF%" }{TEXT -1 360 ". The order in which we \+ add the numbers is immaterial. Starting at the left end of each sum we could add the first number 1 in the upper sum to the first number 199 in the lower sum to make 200, the 2nd number 3 in the upper sum to th e 2nd number 197 in the lower sum and so on until we reach the last nu mber 199 in the upper sum to add to the lower number 1. " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 733 117 117 {PLOTDATA 2 "6^o-%%TEXTG 6$7$$\"\"\"\"\"!F'Q\"16\"-F$6$7$$\"\"#F)F'Q\"3F+-F$6$7$$\"\"$F)F'Q\"5F +-F$6$7$$\"\"%F)F'Q\"7F+-F$6$7$$\"\"(F)F'Q$193F+-F$6$7$$\"\")F)F'Q$195 F+-F$6$7$$\"\"*F)F'Q$197F+-F$6$7$$\"#5F)F'Q$199F+-F$6$7$$\"#:!\"\"F'Q \"+F+-F$6$7$$\"#DFenF'Ffn-F$6$7$$\"#NFenF'Ffn-F$6$7$$\"#XFenF'Ffn-F$6$ 7$$\"#lFenF'Ffn-F$6$7$$\"#vFenF'Ffn-F$6$7$$\"#&)FenF'Ffn-F$6$7$$\"#&*F enF'Ffn-F$6$7$F'$F)F)FU-F$6$7$F/F]qFO-F$6$7$F5F]qFI-F$6$7$F;F]qFC-F$6$ 7$FAF]qF=-F$6$7$FGF]qF7-F$6$7$FMF]qF1-F$6$7$FSF]qF*-F$6$7$FYF]qFfn-F$6 $7$FjnF]qFfn-F$6$7$F_oF]qFfn-F$6$7$FdoF]qFfn-F$6$7$FioF]qFfn-F$6$7$F^p F]qFfn-F$6$7$FcpF]qFfn-F$6$7$FhpF]qFfn-F$6$7$F'$FenF)Q$200F+-F$6$7$F/F ^tF_t-F$6$7$F5F^tF_t-F$6$7$F;F^tF_t-F$6$7$FAF^tF_t-F$6$7$FGF^tF_t-F$6$ 7$FMF^tF_t-F$6$7$FSF^tF_t-F$6$7$FYF^tFfn-F$6$7$FjnF^tFfn-F$6$7$F_oF^tF fn-F$6$7$FdoF^tFfn-F$6$7$FioF^tFfn-F$6$7$F^pF^tFfn-F$6$7$FcpF^tFfn-F$6 $7$FhpF^tFfn-F$6$7$F]qF^tQ\"2F+-F$6%7$$\"#bFenF'Q(.~~.~~.F+-%%FONTG6%% &TIMESG%%BOLDGFT-F$6%7$FdwF]qFfwFgw-F$6%7$FdwF^tFfwFgw-F$6%7$$FF%\"$&>F%\"$(>F%\"$*>F%\"&++\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Sigma nota tion for series " }}{PARA 0 "" 0 "" {TEXT -1 37 "The sum of the first \+ 100 odd numbers " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1 +3+5+7+` . . . `+193+195+197+199" "6#,4\"\"\"F$\"\"$F$\"\"&F$\"\"(F$%( ~.~.~.~GF$\"$$>F$\"$&>F$\"$(>F$\"$*>F$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 98 "considered in the previous section, is the sum of th e terms of the (finite) sequence of numbers: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1,3,5,7,` . . . `,193,195,197,199" "6+ \"\"\"\"\"$\"\"&\"\"(%(~.~.~.~G\"$$>\"$&>\"$(>\"$*>" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "A term of this sequence has the general form " }{XPPEDIT 18 0 "a[i]=2*i-1" "6#/&%\"aG6#%\"iG,&*&\"\"#\"\"\"F' F+F+F+!\"\"" }{TEXT -1 18 ", where the index " }{TEXT 269 1 "i" } {TEXT -1 49 " gives the position of the term in the sequence. " }} {PARA 0 "" 0 "" {TEXT -1 15 "The expression " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[i],i = 1 .. `i =`*n);" "6#-%$SumG 6$&%\"aG6#%\"iG/F);\"\"\"*&%$i~=GF,%\"nGF," }{TEXT -1 13 " or simply \+ " }{XPPEDIT 18 0 "Sum(a[i],i = 1 .. n)" "6#-%$SumG6$&%\"aG6#%\"iG/F); \"\"\"%\"nG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 24 "is used to denote a sum " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[1 ]+a[2]+a[3]+` . . . `+a[i]+` . . . `+a[n-1]+a[n]" "6#,2&%\"aG6#\"\"\"F '&F%6#\"\"#F'&F%6#\"\"$F'%(~.~.~.~GF'&F%6#%\"iGF'F.F'&F%6#,&%\"nGF'F'! \"\"F'&F%6#F5F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 49 "of th e members of an underlying finite sequence: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[1],a[2],a[3],` . . .`,a[i],` . . .`,a [n-1],a[n]" "6*&%\"aG6#\"\"\"&F$6#\"\"#&F$6#\"\"$%'~.~.~.G&F$6#%\"iGF- &F$6#,&%\"nGF&F&!\"\"&F$6#F4" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "The symbol " }{XPPEDIT 18 0 "Sigma" "6#%&SigmaG" }{TEXT -1 71 " is the (upper case) Greek letter sigma, standing for the word \"sum\". " }}{PARA 0 "" 0 "" {TEXT -1 12 "The notation" }{XPPEDIT 18 0 "Sum(a[i],i = 1 .. n)" "6#-%$SumG6$&%\"aG6#%\"iG/F);\"\"\"%\"nG" } {TEXT -1 30 " may be read as \"the sum from " }{TEXT 272 1 "i" }{TEXT -1 13 " equals 1 to " }{TEXT 271 1 "n" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "a[i]" "6#&%\"aG6#%\"iG" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 32 "Such a sum is called a (finite) " }{TEXT 259 6 "series" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "For example, the sum of the first 100 odd numbers is: " } }{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(2*i-1,i = 1 .. 1 00) = 1+3+5+7+` . . . `+193+195+197+199;" "6#/-%$SumG6$,&*&\"\"#\"\"\" %\"iGF*F*F*!\"\"/F+;F*\"$+\",4F*F*\"\"$F*\"\"&F*\"\"(F*%(~.~.~.~GF*\"$ $>F*\"$&>F*\"$(>F*\"$*>F*" }{TEXT -1 2 ". " }}{PARA 258 "" 0 "" {TEXT -1 40 "In the previous section we showed that: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(2*i-1,i = 1 .. 100)=10000" "6#/-%$S umG6$,&*&\"\"#\"\"\"%\"iGF*F*F*!\"\"/F+;F*\"$+\"\"&++\"" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 21 "For another example, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(i^2,i=1..6)=1^2+2^2+3^2+4^2 +5^2+6^2" "6#/-%$SumG6$*$%\"iG\"\"#/F(;\"\"\"\"\"',.*$F,F)F,*$F)F)F,*$ \"\"$F)F,*$\"\"%F)F,*$\"\"&F)F,*$F-F)F," }{TEXT -1 2 ". " }{TEXT 270 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "This is the sum of \+ the squares of the integers from 1 to 6. " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(i^2,i = 1 .. 6) = 1+4+9+16+25+36;" "6#/-% $SumG6$*$%\"iG\"\"#/F(;\"\"\"\"\"',.F,F,\"\"%F,\"\"*F,\"#;F,\"#DF,\"#O F," }{XPPEDIT 18 0 "`` = 91;" "6#/%!G\"#\"*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The starting va lue for the index given below the " }{XPPEDIT 18 0 "Sigma;" "6#%&Sigma G" }{TEXT -1 153 " symbol may be an integer different from but the ind ex must still increase by 1 in moving from one terrm of the next in th e underlying finite sequence. " }}{PARA 0 "" 0 "" {TEXT -1 13 "For ex ample, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/k,k \+ = 3 .. 8) = 1/3+1/4+1/5+1/6+1/7+1/8;" "6#/-%$SumG6$*&\"\"\"F(%\"kG!\" \"/F);\"\"$\"\"),.*&F(F(F-F*F(*&F(F(\"\"%F*F(*&F(F(\"\"&F*F(*&F(F(\"\" 'F*F(*&F(F(\"\"(F*F(*&F(F(F.F*F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Sum(2*i-1,i= 1..100)=sum(2*i-1,i=1..100);\nSum(i^2,i=1..6)=sum(i^2,i=1..6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$,&*&\"\"#\"\"\"%\"iGF*F*F*! \"\"/F+;F*\"$+\"\"&++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$* $)%\"iG\"\"#\"\"\"/F);F+\"\"'\"#\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "The sum of an arithmetic series " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 164 "A formula f or the sum of a finite arithmetic series can be found by the following the same idea as that used previously to find the sum of a series of \+ odd numbers. " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n]=Sum(a[i],i=1..n)" "6#/&%\"SG6#% \"nG-%$SumG6$&%\"aG6#%\"iG/F.;\"\"\"F'" }{XPPEDIT 18 0 "``=a[1]+a[2]+` . . . `+a[n]" "6#/%!G,*&%\"aG6#\"\"\"F)&F'6#\"\"#F)%(~.~.~.~GF)&F'6#% \"nGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "be a finite ari thmetic series with first term " }{XPPEDIT 18 0 "a[1] = c;" "6#/&%\"aG 6#\"\"\"%\"cG" }{TEXT -1 23 " and common difference " }{TEXT 273 1 "d " }{TEXT -1 7 ". Then " }{XPPEDIT 18 0 "a[i] = c+(i-1)*d;" "6#/&%\"aG6 #%\"iG,&%\"cG\"\"\"*&,&F'F*F*!\"\"F*%\"dGF*F*" }{TEXT -1 7 " for " } {XPPEDIT 18 0 "i=1,2,` . . . `,n" "6&/%\"iG\"\"\"\"\"#%(~.~.~.~G%\"nG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "We have: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n] = ` `*c*` \+ `+` `*``(c*` `+` `*d)*` `+` `*``(c*` `+` `*2*d)*` `+` . \+ . . `+``(c+(n-3)*d)+``(c+(n-2)*d)+``(c+(n-1)*d);" "6#/&%\"SG6#%\"nG,0* (%)~~~~~~~~G\"\"\"%\"cGF+F*F+F+*(%\"~GF+-%!G6#,&*&F,F+%$~~~GF+F+*&F4F+ %\"dGF+F+F+F4F+F+*(F.F+-F06#,&*&F,F+F4F+F+*(F4F+\"\"#F+F6F+F+F+F4F+F+% (~.~.~.~GF+-F06#,&F,F+*&,&F'F+\"\"$!\"\"F+F6F+F+F+-F06#,&F,F+*&,&F'F+F =FEF+F6F+F+F+-F06#,&F,F+*&,&F'F+F+FEF+F6F+F+F+" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "S[n] = ``(c+(n-1)*d)+``(c+(n-2)*d)+``(c+(n-3)*d)+` . . \+ . `+` `*``(c*` `+` `*2*d)*` `+` `*``(c*` `+` `*d)*` `+` \+ `*c*` `;" "6#/&%\"SG6#%\"nG,0-%!G6#,&%\"cG\"\"\"*&,&F'F.F. !\"\"F.%\"dGF.F.F.-F*6#,&F-F.*&,&F'F.\"\"#F1F.F2F.F.F.-F*6#,&F-F.*&,&F 'F.\"\"$F1F.F2F.F.F.%(~.~.~.~GF.*(%\"~GF.-F*6#,&*&F-F.%$~~~GF.F.*(FFF. F8F.F2F.F.F.FFF.F.*(FAF.-F*6#,&*&F-F.FFF.F.*&FFF.F2F.F.F.FFF.F.*(%)~~~ ~~~~~GF.F-F.FOF.F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 44 "Adding the terms vertically in pairs give s: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*S[n]=(2*c+(n -1)*d)*n" "6#/*&\"\"#\"\"\"&%\"SG6#%\"nGF&*&,&*&F%F&%\"cGF&F&*&,&F*F&F &!\"\"F&%\"dGF&F&F&F*F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n]=( 2*c+(n-1)*d)*n/2" "6#/&%\"SG6#%\"nG*(,&*&\"\"#\"\"\"%\"cGF,F,*&,&F'F,F ,!\"\"F,%\"dGF,F,F,F'F,F+F0" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 275 13 "_____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Notice that the expression " }{XPPEDIT 18 0 "(2*c+(n-1)*d)" "6#,&*&\"\"#\"\"\"%\"cGF& F&*&,&%\"nGF&F&!\"\"F&%\"dGF&F&" }{TEXT -1 90 " which is obtained by a dding each successive \"vertical pair\" is the sum of the first term \+ " }{XPPEDIT 18 0 "a[1]=c" "6#/&%\"aG6#\"\"\"%\"cG" }{TEXT -1 19 " and \+ the last term " }{XPPEDIT 18 0 "a[n]=c+(n-1)*d" "6#/&%\"aG6#%\"nG,&%\" cG\"\"\"*&,&F'F*F*!\"\"F*%\"dGF*F*" }{TEXT -1 67 ". Hence an alternati ve formula for the sum of an arithmetic series " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n]=Sum(a[i],i=1..n)" "6#/&%\"SG6#%\"n G-%$SumG6$&%\"aG6#%\"iG/F.;\"\"\"F'" }{XPPEDIT 18 0 "``=a[1]+a[2]+` . \+ . . `+a[n]" "6#/%!G,*&%\"aG6#\"\"\"F)&F'6#\"\"#F)%(~.~.~.~GF)&F'6#%\"n GF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "that does not invo lve the common difference " }{TEXT 274 1 "d" }{TEXT -1 4 " is " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n]=(a[1]+a[n])*n/2 " "6#/&%\"SG6#%\"nG*(,&&%\"aG6#\"\"\"F-&F+6#F'F-F-F'F-\"\"#!\"\"" } {TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 276 9 "______ ___" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "Observing that the expression " }{XPPEDIT 18 0 "(a[1]+a[n])/2;" "6#*&,&&%\"aG6#\"\"\"F(& F&6#%\"nGF(F(\"\"#!\"\"" }{TEXT -1 81 " is the average of the first an d last terms, the sum of an arithmetic series is: " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{TEXT 259 65 "the average of the first and last term s times the number of terms" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 159 "For example, the sum of the fi rst 100 odd numbers is the average of the first odd number 1 and the l ast odd number 199, namely 100, times 100, that is, 10000. " }}{PARA 0 "" 0 "" {TEXT -1 38 "The sum of the first 100 even numbers " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2+4+6+8+` . . . `+198 +200" "6#,0\"\"#\"\"\"\"\"%F%\"\"'F%\"\")F%%(~.~.~.~GF%\"$)>F%\"$+#F% " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``((2+200)/2)*100=10100" "6#/*&-%!G6 #*&,&\"\"#\"\"\"\"$+#F+F+F*!\"\"F+\"$+\"F+\"&+,\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Sum(2*i,i = 1 .. 100) = 10100" "6#/-%$SumG6$*&\"\"# \"\"\"%\"iGF)/F*;F)\"$+\"\"&+,\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Sum(2*i,i=1. .100)=sum(2*i,i=1..100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$ ,$*&\"\"#\"\"\"%\"iGF*F*/F+;F*\"$+\"\"&+,\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Geometric sequences " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 23 "Consider t he sequence: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1,1/ 2,1/4,1/8,1/16,1/32,1/64,1/256,` . . . `;" "6+\"\"\"*&F#F#\"\"#!\"\"*& F#F#\"\"%F&*&F#F#\"\")F&*&F#F#\"#;F&*&F#F#\"#KF&*&F#F#\"#kF&*&F#F#\"$c #F&%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "with ge neral term " }{XPPEDIT 18 0 "a[n]=1/2^(n-1)" "6#/&%\"aG6#%\"nG*&\"\"\" F))\"\"#,&F'F)F)!\"\"F-" }{TEXT -1 46 ", and also defined by the recur sion formulas: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PI ECEWISE([a[1]=1,``],[a[n+1]=a[n]*``(1/2),n = 1*`,`*2*`,`*3*`, . . . `] )" "6#-%*PIECEWISEG6$7$/&%\"aG6#\"\"\"F+%!G7$/&F)6#,&%\"nGF+F+F+*&&F)6 #F2F+-F,6#*&F+F+\"\"#!\"\"F+/F2*.F+F+%\",GF+F9F+F=F+\"\"$F+%),~.~.~.~G F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 64 "In general, a sequence with a recursive definition of t he form: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWIS E([a[1] = c, ``],[a[n+1] = a[n]*`.`*r, n = 1*`,`*2*`,`*3*`, . . . `]); " "6#-%*PIECEWISEG6$7$/&%\"aG6#\"\"\"%\"cG%!G7$/&F)6#,&%\"nGF+F+F+*(&F )6#F3F+%\".GF+%\"rGF+/F3*.F+F+%\",GF+\"\"#F+F;F+\"\"$F+%),~.~.~.~GF+" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 277 1 "c" }{TEXT -1 5 " and " }{TEXT 278 1 "r" }{TEXT -1 31 " are fixed (real) numbers with " }{XPPEDIT 18 0 "r<> 0" "6#0%\"rG\"\"!" }{TEXT -1 14 ", is called a " }{TEXT 259 18 "geomet ric sequence" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {TEXT 280 1 "c" }{TEXT -1 40 " is the first term of the sequence, and \+ " }{TEXT 279 1 "r" }{TEXT -1 73 " is the ratio between successive term s of the sequence and is called the " }{TEXT 259 12 "common ratio" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 63 "Such a sequence can als o be described by means of the formula: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[n] = c*r^(n-1);" "6#/&%\"aG6#%\"nG*&%\"cG\" \"\")%\"rG,&F'F*F*!\"\"F*" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 281 8 "________" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "Hence the sequence is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c,c*r,c*r^2,c*r^3,` . . . `,c*r^(n-1),` . . . `;" " 6)%\"cG*&F#\"\"\"%\"rGF%*&F#F%*$F&\"\"#F%*&F#F%*$F&\"\"$F%%(~.~.~.~G*& F#F%)F&,&%\"nGF%F%!\"\"F%F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "The geometric sequence with fir st term 2 and common ratio 10 is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "2,20,200,2000,20000,` . . . `" "6(\"\"#\"#?\"$+#\"%+ ?\"&++#%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 "The geometric sequence with first term 3 and common ratio " }{XPPEDIT 18 0 "1/10" "6#*&\"\"\"F$\"#5!\"\"" }{TEXT -1 5 " is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3,3/10,3/100,3/1000,3/10000,` . . . \+ `" "6(\"\"$*&F#\"\"\"\"#5!\"\"*&F#F%\"$+\"F'*&F#F%\"%+5F'*&F#F%\"&++\" F'%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 116 "From th ese two examples we see that the terms can grow in magnitude very rapi dly, or decrease in magnitude rapidly. " }}{PARA 0 "" 0 "" {TEXT -1 76 "If the common ratio is close to 1, the magnitude does not change s o rapidly." }}{PARA 0 "" 0 "" {TEXT -1 58 "The geometric sequence with first term 1 and common ratio " }{XPPEDIT 18 0 "101/100=1.01" "6#/*& \"$,\"\"\"\"\"$+\"!\"\"-%&FloatG6$F%!\"#" }{TEXT -1 5 " is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "101/100,10201/10000,1030301 /1000000,104060401/100000000,10510100501/10000000000,1061520150601/100 0000000000,107213535210701/100000000000000,10828567056280801/100000000 00000000,` . . . `;" "6+*&\"$,\"\"\"\"\"$+\"!\"\"*&\"&,-\"F%\"&++\"F'* &\"(,..\"F%\"(+++\"F'*&\"*,/1/\"F%\"*++++\"F'*&\",,0550\"F%\",+++++\"F '*&\".,1:?:1\"F%\".++++++\"F'*&\"0,2@NN@2\"F%\"0+++++++\"F'*&\"2,3Gcqc G3\"F%\"2++++++++\"F'%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Converting to decimals an d rounding to 5 digits gives: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1.0100, 1.0201, 1.0303, 1.0406, 1.0510, 1.0615, 1.0721, 1.0829,` . . . `" "6+-%&FloatG6$\"&+,\"!\"%-F$6$\"&,-\"F'-F$6$\"&..\" F'-F$6$\"&1/\"F'-F$6$\"&50\"F'-F$6$\"&:1\"F'-F$6$\"&@2\"F'-F$6$\"&H3\" F'%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "seq((101/100)^n,n=1..8);\nev alf[5](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*#\"$,\"\"$+\"#\"&,-\"\"& ++\"#\"(,..\"\"(+++\"#\"*,/1/\"\"*++++\"#\",,0550\"\",+++++\"#\".,1:?: 1\"\".++++++\"#\"0,2@NN@2\"\"0+++++++\"#\"2,3GcqcG3\"\"2++++++++\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*$\"&+,\"!\"%$\"&,-\"F%$\"&..\"F%$\"&1/ \"F%$\"&50\"F%$\"&:1\"F%$\"&@2\"F%$\"&H3\"F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "The sum of a finite geometric series " }}{PARA 0 "" 0 "" {TEXT -1 40 "Consider the (finite) geometric series: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2+1/(2^2)+1/(2 ^3)+1/(2^4)+1/(2^5)+1/(2^6)+1/(2^7)+1/(2^8);" "6#,2*&\"\"\"F%\"\"#!\" \"F%*&F%F%*$F&F&F'F%*&F%F%*$F&\"\"$F'F%*&F%F%*$F&\"\"%F'F%*&F%F%*$F&\" \"&F'F%*&F%F%*$F&\"\"'F'F%*&F%F%*$F&\"\"(F'F%*&F%F%*$F&\"\")F'F%" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/ 256" "6#/%!G,2*&\"\"\"F'\"\"#!\"\"F'*&F'F'\"\"%F)F'*&F'F'\"\")F)F'*&F' F'\"#;F)F'*&F'F'\"#KF)F'*&F'F'\"#kF)F'*&F'F'\"$G\"F)F'*&F'F'\"$c#F)F' " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "By accumulating successive terms we generate the followin g sequence of " }{TEXT 259 12 "partial sums" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2,3/4,7/8,15/16,31/32,63/ 64,127/128,255/256" "6**&\"\"\"F$\"\"#!\"\"*&\"\"$F$\"\"%F&*&\"\"(F$\" \")F&*&\"#:F$\"#;F&*&\"#JF$\"#KF&*&\"#jF$\"#kF&*&\"$F\"F$\"$G\"F&*&\"$ b#F$\"$c#F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The terms of this sequence are approaching 1. T he " }{TEXT 282 1 "n" }{TEXT -1 28 " th term of the sequence is " } {XPPEDIT 18 0 "(2^n-1)/2^n=1-1/2^n" "6#/*&,&)\"\"#%\"nG\"\"\"F)!\"\"F) )F'F(F*,&F)F)*&F)F))F'F(F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 87 "This is an example of a more general formula for the sum \+ of a finite geometric series: " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n] = a+a*r+a*r^2+a* r^3+` . . .`+a*r^(n-1);" "6#/&%\"SG6#%\"nG,.%\"aG\"\"\"*&F)F*%\"rGF*F* *&F)F**$F,\"\"#F*F**&F)F**$F,\"\"$F*F*%'~.~.~.GF**&F)F*)F,,&F'F*F*!\" \"F*F*" }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 3 "In " }{XPPEDIT 18 0 "Sigma" "6#%&SigmaG" }{TEXT -1 10 " notation " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n] = Sum(a*r^(i-1), i = 1 .. n);" "6#/&%\"SG6#%\"nG-%$SumG6$*&%\"aG\"\"\")%\"rG,&%\"iGF-F- !\"\"F-/F1;F-F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "Now" } }{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r*S[n] = a*r+a*r^2+a *r^3+a*r^4+` . . . `+a*r^(n-1)+a*r^n;" "6#/*&%\"rG\"\"\"&%\"SG6#%\"nGF &,0*&%\"aGF&F%F&F&*&F-F&*$F%\"\"#F&F&*&F-F&*$F%\"\"$F&F&*&F-F&*$F%\"\" %F&F&%(~.~.~.~GF&*&F-F&)F%,&F*F&F&!\"\"F&F&*&F-F&)F%F*F&F&" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 37 "The sum in (i) is \+ obtained by adding " }{TEXT 283 1 "a" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "a*r+a*r^2+a*r^3+` . . .`+a*r^(n-1)" "6#,,*&%\"aG\"\"\"%\"rGF&F&*&F% F&*$F'\"\"#F&F&*&F%F&*$F'\"\"$F&F&%'~.~.~.GF&*&F%F&)F',&%\"nGF&F&!\"\" F&F&" }{TEXT -1 43 ", while the sum (ii) is obtained by adding " } {XPPEDIT 18 0 "a*r^n" "6#*&%\"aG\"\"\")%\"rG%\"nGF%" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 43 "Hence, by subtracting (ii) from (i) we h ave" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n]-r*S[n] = \+ a-a*r^n;" "6#/,&&%\"SG6#%\"nG\"\"\"*&%\"rGF)&F&6#F(F)!\"\",&%\"aGF)*&F 0F))F+F(F)F." }{TEXT -1 3 "., " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1-r)*S[n]=a*(1- r^n)" "6#/*&,&\"\"\"F&%\"rG!\"\"F&&%\"SG6#%\"nGF&*&%\"aGF&,&F&F&)F'F,F (F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 15 "It follows that" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "S[n] = a*(1-r^n)/(1-r);" "6#/&%\"SG6#%\"nG*(%\"aG\"\"\" ,&F*F*)%\"rGF'!\"\"F*,&F*F*F-F.F." }{TEXT -1 16 " ------- (iii). " }} {PARA 0 "" 0 "" {TEXT -1 47 "The expression (iii) is convenient to use when " }{TEXT 284 1 "r" }{TEXT -1 36 " is less than 1, since in this \+ case " }{XPPEDIT 18 0 "1-r^n" "6#,&\"\"\"F$)%\"rG%\"nG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "1-r" "6#,&\"\"\"F$%\"rG!\"\"" }{TEXT -1 20 " are both positive. " }}{PARA 0 "" 0 "" {TEXT -1 20 "If the common ratio " }{TEXT 319 1 "r" }{TEXT -1 42 " is greater than 1, the altern ative form: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n] \+ = a*(r^n-1)/(r-1);" "6#/&%\"SG6#%\"nG*(%\"aG\"\"\",&)%\"rGF'F*F*!\"\"F *,&F-F*F*F.F." }{TEXT -1 15 " ------- (iv), " }}{PARA 0 "" 0 "" {TEXT -1 13 "may be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 83 "The formula (iii) can be used to obtain a formula for s um of the geometric series: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(1/(2^i),i = 1 .. n) = 1/2+1/(2^2)+1/(2^3)+` . . . ` +1/(2^n);" "6#/-%$SumG6$*&\"\"\"F()\"\"#%\"iG!\"\"/F+;F(%\"nG,,*&F(F(F *F,F(*&F(F(*$F*F*F,F(*&F(F(*$F*\"\"$F,F(%(~.~.~.~GF(*&F(F()F*F/F,F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "In this case, the first term is " }{XPPEDIT 18 0 "a=1/2" "6#/%\"aG*&\"\"\"F&\"\"#!\"\"" } {TEXT -1 25 " and the common ratio is " }{XPPEDIT 18 0 "r=1/2" "6#/%\" rG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 7 " also. " }}{PARA 0 "" 0 "" {TEXT -1 26 "The sum of the series is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(1/2)*(1-1/2^n)/(1-1/2)=(1-1/2^n)" "6#/*(-%!G6#*&\" \"\"F)\"\"#!\"\"F),&F)F)*&F)F))F*%\"nGF+F+F),&F)F)*&F)F)F*F+F+F+,&F)F) *&F)F))F*F/F+F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 59 "which checks with the result at the start of this section. " }}{PARA 0 "" 0 "" {TEXT -1 51 "For another example consider the geometric series: \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(2*(4/3)^(i-1) ,i = 1 .. 7) = 2+8/3+32/9+128/27+512/81+2048/243+8192/729;" "6#/-%$Sum G6$*&\"\"#\"\"\")*&\"\"%F)\"\"$!\"\",&%\"iGF)F)F.F)/F0;F)\"\"(,0F(F)*& \"\")F)F-F.F)*&\"#KF)\"\"*F.F)*&\"$G\"F)\"#FF.F)*&\"$7&F)\"#\")F.F)*& \"%[?F)\"$V#F.F)*&\"%#>)F)\"$H(F.F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 43 "The first term of this geometric series is " }{XPPEDIT 18 0 "a = 2;" "6#/%\"aG\"\"#" }{TEXT -1 22 ", the common ratio is " } {XPPEDIT 18 0 "r=4/3" "6#/%\"rG*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT -1 28 " and the number of terms is " }{XPPEDIT 18 0 "n = 7;" "6#/%\"nG\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "Hence, using the formu la (iv), the sum of the series is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "2*((4/3)^7-1)/(4/3-1) = 2*(16384/2187-1)/``(1/3);" " 6#/*(\"\"#\"\"\",&*$*&\"\"%F&\"\"$!\"\"\"\"(F&F&F,F&,&*&F*F&F+F,F&F&F, F,*(F%F&,&*&\"&%Q;F&\"%(=#F,F&F&F,F&-%!G6#*&F&F&F+F,F," }{XPPEDIT 18 0 "`` = 6*``((16384-2187)/2187);" "6#/%!G*&\"\"'\"\"\"-F$6#*&,&\"&%Q;F '\"%(=#!\"\"F'F-F.F'" }{XPPEDIT 18 0 "`` = 6*14197/2187;" "6#/%!G*(\" \"'\"\"\"\"&(>9F'\"%(=#!\"\"" }{XPPEDIT 18 0 "`` = 2*14197/729;" "6#/% !G*(\"\"#\"\"\"\"&(>9F'\"$H(!\"\"" }{XPPEDIT 18 0 "`` = 28394/729;" "6 #/%!G*&\"&%RG\"\"\"\"$H(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " } {MPLTEXT 1 0 3 "sum" }{TEXT -1 62 " applies the formula (iv) for summi ng finite geometric series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Sum(a*r^(i-1),i = 1 .. n);\n``=simp lify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&%\"aG\" \"\")%\"rG,&%\"iGF(F(!\"\"F(/F,;F(%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(%\"aG\"\"\",&)%\"rG%\"nGF'F'!\"\"F',&F*F'F'F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Su m(2*(4/3)^(i-1),i=1..7)=sum(2*(4/3)^(i-1),i=1..7);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$SumG6$,$*&\"\"#\"\"\")#\"\"%\"\"$,&%\"iGF*F*!\"\" F*F*/F0;F*\"\"(#\"&%RG\"$H(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 82 "A geometric series connected with compound interest on \+ regular periodic deposits " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 23 "Suppose that an amount " }{TEXT 303 1 "A" }{TEXT -1 97 " dollars is added to a savings account annuall y, and that the interest rate on the investment is " }{TEXT 307 1 "p" }{TEXT -1 29 " percent compouned annually. " }}{PARA 0 "" 0 "" {TEXT -1 56 "Such a sequence of equal periodic payments is called an " } {TEXT 259 7 "annuity" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 " Then after 1 year the original amount earns " }{XPPEDIT 18 0 "A*``(p/1 00)" "6#*&%\"AG\"\"\"-%!G6#*&%\"pGF%\"$+\"!\"\"F%" }{TEXT -1 31 " doll ars interest and grows to " }{XPPEDIT 18 0 "A+A*``(p/100)=A*(1+p/100) " "6#/,&%\"AG\"\"\"*&F%F&-%!G6#*&%\"pGF&\"$+\"!\"\"F&F&*&F%F&,&F&F&*&F ,F&F-F.F&F&" }{TEXT -1 10 " dollars. " }}{PARA 0 "" 0 "" {TEXT -1 34 " At that time an additional amount " }{TEXT 304 1 "A" }{TEXT -1 39 " is invested so that the account holds " }{XPPEDIT 18 0 "A[1]=A+A*(1+p/10 0)" "6#/&%\"AG6#\"\"\",&F%F'*&F%F',&F'F'*&%\"pGF'\"$+\"!\"\"F'F'F'" } {TEXT -1 10 " dollars. " }}{PARA 0 "" 0 "" {TEXT -1 63 "This amount ea rns interest, so after a second year it grows to " }}{PARA 257 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "A[1]*(1+p/100)=(A+A*(1+p/100))*(1+p/ 100)" "6#/*&&%\"AG6#\"\"\"F(,&F(F(*&%\"pGF(\"$+\"!\"\"F(F(*&,&F&F(*&F& F(,&F(F(*&F+F(F,F-F(F(F(F(,&F(F(*&F+F(F,F-F(F(" }{XPPEDIT 18 0 "``=A*( 1+p/100)+A*(1+p/100)^2" "6#/%!G,&*&%\"AG\"\"\",&F(F(*&%\"pGF(\"$+\"!\" \"F(F(F(*&F'F(*$,&F(F(*&F+F(F,F-F(\"\"#F(F(" }{TEXT -1 10 " dollars. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Since another " }{TEXT 305 1 "A" }{TEXT -1 60 " dollars is added to the acc ount at this time the total is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "A+A*(1+p/100)+A*(1+p/100)^2" "6#,(%\"AG\"\"\"*&F$F%,&F% F%*&%\"pGF%\"$+\"!\"\"F%F%F%*&F$F%*$,&F%F%*&F)F%F*F+F%\"\"#F%F%" } {TEXT -1 10 " dollars. " }}{PARA 0 "" 0 "" {TEXT -1 70 "The amount in \+ the account after 3 years (before the new deposit) is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A*(1+p/100)+A*(1+p/100)^2+A*(1+ p/100)^3" "6#,(*&%\"AG\"\"\",&F&F&*&%\"pGF&\"$+\"!\"\"F&F&F&*&F%F&*$,& F&F&*&F)F&F*F+F&\"\"#F&F&*&F%F&*$,&F&F&*&F)F&F*F+F&\"\"$F&F&" }{TEXT -1 10 " dollars. " }}{PARA 0 "" 0 "" {TEXT -1 6 "After " }{TEXT 308 1 "n" }{TEXT -1 34 " years the amount accumulated is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A*(1+p/100)+A*(1+p/100)^2+` . . . \+ `+A*(1+p/100)^n" "6#,**&%\"AG\"\"\",&F&F&*&%\"pGF&\"$+\"!\"\"F&F&F&*&F %F&*$,&F&F&*&F)F&F*F+F&\"\"#F&F&%(~.~.~.~GF&*&F%F&),&F&F&*&F)F&F*F+F&% \"nGF&F&" }{TEXT -1 10 " dollars. " }}{PARA 0 "" 0 "" {TEXT -1 47 "Not e that this is a finite geometric series of " }{TEXT 306 1 "n" }{TEXT -1 23 " terms with first term " }{XPPEDIT 18 0 "A*(1+p/100)" "6#*&%\"A G\"\"\",&F%F%*&%\"pGF%\"$+\"!\"\"F%F%" }{TEXT -1 19 ", and common rati o " }{XPPEDIT 18 0 "(1+p/100)" "6#,&\"\"\"F$*&%\"pGF$\"$+\"!\"\"F$" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 74 "Using the formula for t he sum of a finite geometric series this total is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A*(1+p/100)*((1+p/100)^n-1)/(``(1+p/ 100)-1);" "6#**%\"AG\"\"\",&F%F%*&%\"pGF%\"$+\"!\"\"F%F%,&),&F%F%*&F(F %F)F*F%%\"nGF%F%F*F%,&-%!G6#,&F%F%*&F(F%F)F*F%F%F%F*F*" }{TEXT -1 10 " dollars. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "For example, if $3000 dollars is invested annually at 6% compo und interest, the total amount available after 18 years is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3000*``(1.06)*(1.06^18-1)/( 1.06-1) = 98279.98;" "6#/**\"%+I\"\"\"-%!G6#-%&FloatG6$\"$1\"!\"#F&,&* $-F+6$F-F.\"#=F&F&!\"\"F&,&-F+6$F-F.F&F&F4F4-F+6$\"()*z#)*F." }{TEXT -1 10 " dollars. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "For another example, if $1000 is invested annually at 5% compound interest, the total amount available after 40 years is: " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1000*``(1.05)*(1.05^4 0-1)/(1.05-1)=126839.76" "6#/**\"%+5\"\"\"-%!G6#-%&FloatG6$\"$0\"!\"#F &,&*$-F+6$F-F.\"#SF&F&!\"\"F&,&-F+6$F-F.F&F&F4F4-F+6$\")wRo7F." } {TEXT -1 10 " dollars. " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "3000 *(1.06)*((1.06)^18-1)/(0.06):\nevalf[7](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"()*z#)*!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "(1000*(1.05)*((1.05)^40-1)/0.05):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")wRo7!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 64 "The sum of an infinite geometric series (introductory e xample) " }}{PARA 0 "" 0 "" {TEXT -1 35 "The terms of the geometric s eries: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2+4+8+16+3 2+` . . . `" "6#,.\"\"#\"\"\"\"\"%F%\"\")F%\"#;F%\"#KF%%(~.~.~.~GF%" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "with common ratio " } {XPPEDIT 18 0 "r=2" "6#/%\"rG\"\"#" }{TEXT -1 46 ", get larger and lar ger. The sum of the first " }{TEXT 285 1 "n" }{TEXT -1 11 " terms is: \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*(2^n-1)/(2-1)=2 ^(n+1)-2" "6#/*(\"\"#\"\"\",&)F%%\"nGF&F&!\"\"F&,&F%F&F&F*F*,&)F%,&F)F &F&F&F&F%F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "and this i ncreases without bound as " }{TEXT 309 1 "n" }{TEXT -1 12 " increases. " }}{PARA 0 "" 0 "" {TEXT -1 53 "On the other hand, the terms of the \+ geometric series " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " 1/2+1/4+1/8+1/16+1/32+` . . . `" "6#,.*&\"\"\"F%\"\"#!\"\"F%*&F%F%\"\" %F'F%*&F%F%\"\")F'F%*&F%F%\"#;F'F%*&F%F%\"#KF'F%%(~.~.~.~GF%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "with common ration " } {XPPEDIT 18 0 "r=1/2" "6#/%\"rG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 48 ", g et smaller and smaller. The sum of the first " }{TEXT 286 1 "n" } {TEXT -1 10 " terms is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n]=Sum(1/2^i,i=1..n) " "6#/&%\"SG6#%\"nG-%$SumG6$*&\"\"\"F,)\" \"#%\"iG!\"\"/F/;F,F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(1/2)*(1 -1/(2^n))/(1-1/2);" "6#/%!G*(-F$6#*&\"\"\"F)\"\"#!\"\"F),&F)F)*&F)F))F *%\"nGF+F+F),&F)F)*&F)F)F*F+F+F+" }{XPPEDIT 18 0 "`` = 1-1/(2^n)" "6#/ %!G,&\"\"\"F&*&F&F&)\"\"#%\"nG!\"\"F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 49 "and this remains less than 1 no matter how large " } {TEXT 287 1 "n" }{TEXT -1 29 " is. Indeed this sum becomes " }{TEXT 259 23 "progressively closer to" }{TEXT -1 6 " 1 as " }{TEXT 288 1 "n " }{TEXT -1 12 " increases. " }}{PARA 0 "" 0 "" {TEXT -1 40 "This can \+ be seen from the fact the term " }{XPPEDIT 18 0 "1/2^n" "6#*&\"\"\"F$) \"\"#%\"nG!\"\"" }{TEXT -1 19 " in the expression " }{XPPEDIT 18 0 "1- 1/2^n" "6#,&\"\"\"F$*&F$F$)\"\"#%\"nG!\"\"F)" }{TEXT -1 32 " becomes p rogressively smaller. " }}{PARA 0 "" 0 "" {TEXT -1 10 "We write: " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n]->1" "6#f*6#&%\"S G6#%\"nG7\"6$%)operatorG%&arrowG6\"\"\"\"F-F-F-" }{TEXT -1 5 ", as " } {XPPEDIT 18 0 "n->infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)i nfinityGF*F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "For ex ample, the sum of 10 terms is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(1/(2^i),i = 1 .. 10) = 1/2+1/4+1/8+1/16+1/32+1/64+1 /128+1/256+1/512+1/1024;" "6#/-%$SumG6$*&\"\"\"F()\"\"#%\"iG!\"\"/F+;F (\"#5,6*&F(F(F*F,F(*&F(F(\"\"%F,F(*&F(F(\"\")F,F(*&F(F(\"#;F,F(*&F(F( \"#KF,F(*&F(F(\"#kF,F(*&F(F(\"$G\"F,F(*&F(F(\"$c#F,F(*&F(F(\"$7&F,F(*& F(F(\"%C5F,F(" }{XPPEDIT 18 0 "`` = 1023/1024" "6#/%!G*&\"%B5\"\"\"\"% C5!\"\"" }{XPPEDIT 18 0 "``= 0" "6#/%!G\"\"!" }{TEXT -1 13 ".999023437 5, " }}{PARA 0 "" 0 "" {TEXT -1 29 "while the sum of 20 terms is " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/2^i,i=1..20)=1/ 2+1/4+1/8+1/16+1/32+1/64+` . . . `+1/524288+1/1048576" "6#/-%$SumG6$*& \"\"\"F()\"\"#%\"iG!\"\"/F+;F(\"#?,4*&F(F(F*F,F(*&F(F(\"\"%F,F(*&F(F( \"\")F,F(*&F(F(\"#;F,F(*&F(F(\"#KF,F(*&F(F(\"#kF,F(%(~.~.~.~GF(*&F(F( \"')GC&F,F(*&F(F(\"(w&[5F,F(" }{XPPEDIT 18 0 "``=1048575/1048576" "6#/ %!G*&\"(v&[5\"\"\"\"(w&[5!\"\"" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" } {TEXT -1 23 ".99999904632568359375. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We say that the series " }{TEXT 259 9 " converges" }{TEXT -1 17 " to 1 and write: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(S[n],n = infinity) = 1;" "6#/-%&LimitG6 $&%\"SG6#%\"nG/F*%)infinityG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 29 "More simply, we consider the " }{TEXT 259 15 "infinite \+ series" }{TEXT -1 2 ": " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2+1/4+1/8+1/16+1/32+1/64+` . . . `+1/2^n+` . . . `" "6#,4*&\" \"\"F%\"\"#!\"\"F%*&F%F%\"\"%F'F%*&F%F%\"\")F'F%*&F%F%\"#;F'F%*&F%F%\" #KF'F%*&F%F%\"#kF'F%%(~.~.~.~GF%*&F%F%)F&%\"nGF'F%F2F%" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 12 "to have the " }{TEXT 259 3 "sum" } {TEXT -1 13 " 1 and write " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(1/(2^i),i = 1 .. infinity)=1" "6#/-%$SumG6$*&\"\"\" F()\"\"#%\"iG!\"\"/F+;F(%)infinityGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Sum(1/2 ^i,i=1..infinity);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% $SumG6$*&\"\"\"F')\"\"#%\"iG!\"\"/F*;F'%)infinityG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 55 "The sum of an infinite geometric series (general case) \+ " }}{PARA 0 "" 0 "" {TEXT -1 44 "Suppose that the infinite geometric s eries: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a*r^(i -1),i = 1 .. infinity) = a+a*r+a*r^2+a*r^3+` . . . `+a*r^i+` . . . `; " "6#/-%$SumG6$*&%\"aG\"\"\")%\"rG,&%\"iGF)F)!\"\"F)/F-;F)%)infinityG, 0F(F)*&F(F)F+F)F)*&F(F)*$F+\"\"#F)F)*&F(F)*$F+\"\"$F)F)%(~.~.~.~GF)*&F (F))F+F-F)F)F:F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "has a common ratio " }{TEXT 289 1 "r" }{TEXT -1 6 " with " }{XPPEDIT 18 0 " abs(r)<1" "6#2-%$absG6#%\"rG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 11 "The sum of " }{TEXT 290 1 "n" }{TEXT -1 11 " terms is: \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n] = a*(1-r^n)/ (1-r);" "6#/&%\"SG6#%\"nG*(%\"aG\"\"\",&F*F*)%\"rGF'!\"\"F*,&F*F*F-F.F ." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "The expression " } {XPPEDIT 18 0 "r^n" "6#)%\"rG%\"nG" }{TEXT -1 34 " becomes progressive ly smaller as " }{TEXT 291 1 "n" }{TEXT -1 12 " increases. " }}{PARA 0 "" 0 "" {TEXT -1 8 "Even if " }{TEXT 292 1 "r" }{TEXT -1 28 " is clo se to 1 (or close to " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 4 "), " }{XPPEDIT 18 0 "r^n" "6#)%\"rG%\"nG" }{TEXT -1 116 " gets \+ progressively smaller, perhaps rather slowly to start with, but eventu ally it becomes arbitrarily close to 0. " }}{PARA 0 "" 0 "" {TEXT -1 15 "For example, 0." }{XPPEDIT 18 0 "99^10" "6#*$\"#**\"#5" }{TEXT -1 1 " " }{TEXT 293 1 "~" }{TEXT -1 23 " 0.9043820750, while 0." } {XPPEDIT 18 0 "99^100;" "6#*$\"#**\"$+\"" }{TEXT -1 1 " " }{TEXT 294 1 "~" }{TEXT -1 20 " 0.3660323413 and 0." }{XPPEDIT 18 0 "99^1000;" "6 #*$\"#**\"%+5" }{TEXT -1 1 " " }{TEXT 295 1 "~" }{TEXT -1 18 " 0.00004 317124741." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "It follows that, when " }{XPPEDIT 18 0 "abs(r)<1" "6#2-%$absG6# %\"rG\"\"\"" }{TEXT -1 17 ", the expression " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n] = a*(1-r^n)/(1-r);" "6#/&%\"SG6#% \"nG*(%\"aG\"\"\",&F*F*)%\"rGF'!\"\"F*,&F*F*F-F.F." }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 11 "approaches " }{XPPEDIT 18 0 "a/(1-r);" "6 #*&%\"aG\"\"\",&F%F%%\"rG!\"\"F(" }{TEXT -1 11 ", that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a*r^(i-1),i = 1 .. infi nity) = a/(1-r);" "6#/-%$SumG6$*&%\"aG\"\"\")%\"rG,&%\"iGF)F)!\"\"F)/F -;F)%)infinityG*&F(F),&F)F)F+F.F." }{TEXT -1 2 ". " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{TEXT 296 12 "____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 40 "Consider the infinite geometric series: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/3^(i-1),i=1..infinity)=1+1/3 +1/9+1/27+` . . . `" "6#/-%$SumG6$*&\"\"\"F()\"\"$,&%\"iGF(F(!\"\"F-/F ,;F(%)infinityG,,F(F(*&F(F(F*F-F(*&F(F(\"\"*F-F(*&F(F(\"#FF-F(%(~.~.~. ~GF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "The first term i s " }{XPPEDIT 18 0 "a = 1;" "6#/%\"aG\"\"\"" }{TEXT -1 25 " and the co mmon ratio is " }{XPPEDIT 18 0 "r = 1/3;" "6#/%\"rG*&\"\"\"F&\"\"$!\" \"" }{TEXT -1 31 ". Hence the series has the sum " }{XPPEDIT 18 0 "a/( 1-r) = 1/(1-1/3);" "6#/*&%\"aG\"\"\",&F&F&%\"rG!\"\"F)*&F&F&,&F&F&*&F& F&\"\"$F)F)F)" }{XPPEDIT 18 0 "``=3/2" "6#/%!G*&\"\"$\"\"\"\"\"#!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 55 "The convergence is q uite rapid. The sum of 15 terms is " }{XPPEDIT 18 0 "Sum(1/(3^(i-1)),i = 1 .. 15)=7174453/4782969" "6#/-%$SumG6$*&\"\"\"F()\"\"$,&%\"iGF(F(! \"\"F-/F,;F(\"#:*&\"(`W<(F(\"(pHy%F-" }{TEXT -1 1 " " }{TEXT 297 1 "~ " }{TEXT -1 19 " 1.49999989546242. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Sum(1/(3^(i-1)),i=1..infini ty);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\" \"F')\"\"$,&%\"iGF'F'!\"\"F,/F+;F'%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Sum(1/(3^(i-1)),i=1..15);\n``=value(%);\n``=evalf[15](evalf[18 ](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F')\" \"$,&%\"iGF'F'!\"\"F,/F+;F'\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%! G#\"(`W<(\"(pHy%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"0Uia*)**** \\\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 40 "Consider the infinite geometric series: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(5*(4/5)^(i-1),i = 1 .. infinit y) = 5+4+16/5+64/25+256/125+1024/625+` . . . `;" "6#/-%$SumG6$*&\"\"& \"\"\")*&\"\"%F)F(!\"\",&%\"iGF)F)F-F)/F/;F)%)infinityG,0F(F)F,F)*&\"# ;F)F(F-F)*&\"#kF)\"#DF-F)*&\"$c#F)\"$D\"F-F)*&\"%C5F)\"$D'F-F)%(~.~.~. ~GF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "The first term i s " }{XPPEDIT 18 0 "a = 5;" "6#/%\"aG\"\"&" }{TEXT -1 25 " and the com mon ratio is " }{XPPEDIT 18 0 "r = 4/5;" "6#/%\"rG*&\"\"%\"\"\"\"\"&! \"\"" }{TEXT -1 31 ". Hence the series has the sum " }{XPPEDIT 18 0 "a /(1-r) = 5/(1-4/5);" "6#/*&%\"aG\"\"\",&F&F&%\"rG!\"\"F)*&\"\"&F&,&F&F &*&\"\"%F&F+F)F)F)" }{XPPEDIT 18 0 "`` = 25;" "6#/%!G\"#D" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 107 "The convergence is not as fast \+ as in the first example. The sum of 30 terms is approximately 24.96905 150. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The sum of " }{TEXT 300 1 "n" }{TEXT -1 10 " terms is " }{XPPEDIT 18 0 "S[n] = a*(1-r^n)/(1-r);" "6#/&%\"SG6#%\"nG*(%\"aG\"\"\",&F*F*)%\"rG F'!\"\"F*,&F*F*F-F.F." }{XPPEDIT 18 0 "`` = 25*(1-(4/5)^n);" "6#/%!G*& \"#D\"\"\",&F'F')*&\"\"%F'\"\"&!\"\"%\"nGF-F'" }{TEXT -1 65 ", and the first few terms of the sequence of \"partial sums\" are: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "5,9,61/5,369/25,2101/125,11 529/625,` . . . `;" "6)\"\"&\"\"**&\"#h\"\"\"F#!\"\"*&\"$p$F'\"#DF(*& \"%,@F'\"$D\"F(*&\"&H:\"F'\"$D'F(%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "The following p icture illustrates the first few partial sums of this series by plotti ng the points" }{XPPEDIT 18 0 "``(n,S[n]);" "6#-%!G6$%\"nG&%\"SG6#F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "pts := [seq([n,25*(1-(4/5)^n)],n=1..15)]:\nf := x -> 25*(1-(4/5)^x):\np1:=plot([pts$4],style=point,symbol=[circle,dia mond,cross,circle],\n color=[green$3,black],symbolsize=[10$3,12]):\np 2:=plot([f(x),25],x=0..15,color=[COLOR(RGB,.3,.3,.3)$2,navy],linestyle =[2,2,3]):\nplots[display]([p1,p2],view=[0..15,0..26],labels=[`x`,`y`] );" }}{PARA 13 "" 1 "" {GLPLOT2D 569 246 246 {PLOTDATA 2 "6*-%'CURVESG 6&717$$\"\"\"\"\"!$\"\"&F*7$$\"\"#F*$\"\"*F*7$$\"\"$F*$\"3$*********** **>7!#;7$$\"\"%F*$\"3)************fZ\"F77$F+$\"3)***********z!o\"F77$$ \"\"'F*$\"31+++++kW=F77$$\"\"(F*$\"31++++?rv>F77$$\"\")F*$\"37++++'p03 #F77$F0$\"3-+++!obW;#F77$$\"#5F*$\"34+++WXcJAF77$$\"#6F*$\"3')****>N;D &G#F77$$\"#7F*$\"3*)***f\"38?GBF77$$\"#8F*$\"37+!Gl/hDO#F77$$\"#9F*$\" 3/+CAP)[+R#F77$$\"#:F*$\"3-?zxp!R?T#F7-%'COLOURG6&%$RGBG$F*F*$\"*++++ \"!\")F^p-%'SYMBOLG6$%'CIRCLEGFT-%&STYLEG6#%&POINTG-F$6&F&Fjo-Fcp6$%(D IAMONDGFTFfp-F$6&F&Fjo-Fcp6$%&CROSSGFTFfp-F$6&F&-F[p6&F]pF*F*F*-Fcp6$F epFhnFfp-F$6%7S7$F^pF^p7$$\"3')*****\\7t&pK!#=$\"3kSe]7G,f0)H&Fdr$\"3&[_E`N6Nt\"F77$$\"3Y**\\(=-p6j&Fdr$\"3K=F0N 1U)y\"F77$$\"3d*****\\2Mg#fFdr$\"3%*ffK7NtL=F77$$\"35+](=xZ&\\iFdr$\"3 qmsrIi8!)=F77$$\"3;+]i:$4wb'Fdr$\"3Rae*RH:8#>F77$$\"3-++v=#R!zoFdr$\"3 AVu4/zOh>F77$$\"3q+]P4A@urFdr$\"3C.bjtAq&*>F77$$\"3I++Dchf#\\(Fdr$\"3) 3>Fc<(GI?F77$$\"3))**\\(of2L#yFdr$\"3#4/MZ0-P1#F77$$\"3M**\\7yG>6\")Fd r$\"3!\\@c%\\%[34#F77$$\"3w++voo6A%)Fdr$\"3kO/R'Gt#=@F77$$\"3q*****\\x JLu)Fdr$\"3Xi\"f2wwY9#F77$$\"3W++v$*ydd!*Fdr$\"3S:;EA#R(o@F77$$\"3#*** \\(=#F77$$\"35***\\i0A#*p*Fdr$\"30`@Z$pHH@#F77$ $\"3*)****\\2mD+5F7$\"3'p+tp?=D!eB/D #F77$$\"3%**\\(o/Q*>1\"F7$\"39+u6)eWiE#F77$$\"3=++vQ(zS4\"F7$\"3[kW0Te R#G#F77$$\"3***\\(=-,FC6F7$\"3-lW;zCd'H#F77$$\"33+v$4tFe:\"F7$\"3R$))4 %e[S5BF77$$\"3!****\\73\"o'=\"F7$\"3D+AG\"F7$\"3% f\"\\SBz*oN#F77$$\"34+v=(4bMJ\"F7$\"3ek6r1iimBF77$$\"3;++]xlWU8F7$\"3W 4ONN8)\\P#F77$$\"39+]i&3ucP\"F7$\"3#H`UKa:RQ#F77$$\"3\"******\\;$R09F7 $\"3)e-cU6k8R#F77$$\"38+v=-*zqV\"F7$\"3a)f*>],y)R#F77$$\"33+D\"G:3uY\" F7$\"3()>s*pg.aS#F7Feo-%&COLORG6&F]p$F4!\"\"F^alF^al-%*LINESTYLEG6#F/- F$6%7S7$F^p$\"#DF*7$F_rFgal7$FfrFgal7$F[sFgal7$F`sFgal7$FesFgal7$FjsFg al7$F_tFgal7$FdtFgal7$FitFgal7$F^uFgal7$FcuFgal7$FhuFgal7$F]vFgal7$Fbv Fgal7$FgvFgal7$F\\wFgal7$FawFgal7$FfwFgal7$F[xFgal7$F`xFgal7$FexFgal7$ FjxFgal7$F_yFgal7$FdyFgal7$FiyFgal7$F^zFgal7$FczFgal7$FhzFgal7$F][lFga l7$Fb[lFgal7$Fg[lFgal7$F\\\\lFgal7$Fa\\lFgal7$Ff\\lFgal7$F[]lFgal7$F`] lFgal7$Fe]lFgal7$Fj]lFgal7$F_^lFgal7$Fd^lFgal7$Fi^lFgal7$F^_lFgal7$Fc_ lFgal7$Fh_lFgal7$F]`lFgal7$Fb`lFgal7$Fg`lFgal7$FfoFgalF[alF`al-%+AXESL ABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F^pFfo;F^p$\"#EF*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Sum(5*(4/5)^(i-1), i=1..infinity);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Su mG6$,$*&\"\"&\"\"\")#\"\"%F(,&%\"iGF)F)!\"\"F)F)/F.;F)%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"#D" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Sum(5*(4/5)^(i-1),i=1 ..30);\n``=value(%);\n``=evalf[10](evalf[18](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,$*&\"\"&\"\"\")#\"\"%F(,&%\"iGF)F)!\"\"F )F)/F.;F)\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"6\\'o;(36`'p,$ *\"5D19>Y)H!HDP" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+]^!p\\#!\") " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 40 "Consider the infinite geometric series: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum((-1/2)^(i-1),i=1..infinity)=1- 1/2+1/4-1/8+1/16-1/32+` . . . `" "6#/-%$SumG6$),$*&\"\"\"F*\"\"#!\"\"F ,,&%\"iGF*F*F,/F.;F*%)infinityG,0F*F**&F*F*F+F,F,*&F*F*\"\"%F,F**&F*F* \"\")F,F,*&F*F*\"#;F,F**&F*F*\"#KF,F,%(~.~.~.~GF*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 18 "The first term is " }{XPPEDIT 18 0 "a = 1 ;" "6#/%\"aG\"\"\"" }{TEXT -1 25 " and the common ratio is " } {XPPEDIT 18 0 "r = -1/2;" "6#/%\"rG,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 31 ". Hence the series has the sum " }{XPPEDIT 18 0 "a/(1-r) = 1/(1 -(-1/2));" "6#/*&%\"aG\"\"\",&F&F&%\"rG!\"\"F)*&F&F&,&F&F&,$*&F&F&\"\" #F)F)F)F)" }{XPPEDIT 18 0 "`` = 2/3;" "6#/%!G*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "The some of " }{TEXT 298 1 "n" }{TEXT -1 25 " terms of the series is: " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "S[n]=Sum((-1/2)^(i-1),i=1..infinity) " "6#/&%\"SG6#%\"nG-%$SumG6$),$*&\"\"\"F.\"\"#!\"\"F0,&%\"iGF.F.F0/F2; F.%)infinityG" }}{PARA 257 "" 0 "" {TEXT -1 7 " = " }{XPPEDIT 18 0 "(1-(-1/2)^n)/(1-(-1/2)) = 2/3;" "6#/*&,&\"\"\"F&),$*&F&F&\"\"#!\"\" F+%\"nGF+F&,&F&F&,$*&F&F&F*F+F+F+F+*&F*F&\"\"$F+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(1-(-1/2)^n);" "6#-%!G6#,&\"\"\"F'),$*&F'F'\"\"#!\"\" F,%\"nGF," }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "S[n] = PIECEWISE([` `(2/3)*(1-1/(2^n)), ` if n is even`],[``(2/3)*(1+1/(2^n)), ` if n is o dd`])" "6#/&%\"SG6#%\"nG-%*PIECEWISEG6$7$*&-%!G6#*&\"\"#\"\"\"\"\"$!\" \"F2,&F2F2*&F2F2)F1F'F4F4F2%.~if~n~is~evenG7$*&-F.6#*&F1F2F3F4F2,&F2F2 *&F2F2)F1F'F4F2F2%-~if~n~is~oddG" }{TEXT -1 73 " , so that the \"parti al sums\" are alternately greater than and less than " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "The following picture ill ustrates the first few partial sums of this series by plotting the poi nts" }{XPPEDIT 18 0 "``(n,S[n]);" "6#-%!G6$%\"nG&%\"SG6#F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 362 "pts := [seq([n,4/3*(-1/2)^(n+1)+2/3],n=1..8)]:\nf := x -> 2/3+2/3*1/2^x: g := x -> 2/3-2/3*1/2^x:\np1:=plot([pts$4],style= point,symbol=[circle,diamond,cross,circle],\n color=[green$3,black],s ymbolsize=[10$3,12]):\np2:=plot([f(x),g(x),2/3],x=0..10,color=[COLOR(R GB,.3,.3,.3)$2,navy],linestyle=[2,2,3]):\nplots[display]([p1,p2],view= [0..8,-.09..1.2],labels=[`x`,`y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 609 388 388 {PLOTDATA 2 "6+-%'CURVESG6&7*7$$\"\"\"\"\"!F(7$$\"\"#F*$\"3+++ +++++]!#=7$$\"\"$F*$\"3++++++++vF07$$\"\"%F*$\"3+++++++]iF07$$\"\"&F*$ \"3+++++++voF07$$\"\"'F*$\"3++++++]ilF07$$\"\"(F*$\"3++++++v=nF07$$\" \")F*$\"3+++++]iSmF0-%'COLOURG6&%$RGBG$F*F*$\"*++++\"!\")FS-%'SYMBOLG6 $%'CIRCLEG\"#5-%&STYLEG6#%&POINTG-F$6&F&FO-FX6$%(DIAMONDGFenFfn-F$6&F& FO-FX6$%&CROSSGFenFfn-F$6&F&-FP6&FRF*F*F*-FX6$FZ\"#7Ffn-F$6%7W7$FS$\"3 ELLLLLLL8!#<7$$\"3GLLL3x&)*3\"F0$\"3UbE6hk#[G\"Fap7$$\"3emmm;arz@F0$\" 3OF4(Q'*[)R7Fap7$$\"3v***\\7y%*z7$F0$\"3?J$*Q_`Q.7Fap7$$\"3[LL$e9ui2%F 0$\"31?:(HTT#p6Fap7$$\"3z***\\(oMrU^F0$\"3897zu4VL6Fap7$$\"3nmmm\"z_\" 4iF0$\"3ZB*[>%p()F07$$\"33++D1J:w=Fap$\"3m7U/Zus#[)F07$$ \"3oLLL3En$4#Fap$\"3Lb[#>\"obG#)F07$$\"3#pmmT!RE&G#Fap$\"3+M]3[\">V.)F 07$$\"3D+++D.&4]#Fap$\"3$R$f3@>SWyF07$$\"3;+++vB_f\\.\\B!o( F07$$\"33+++v'Hi#HFap$\"3[4n))e'>Pa(F07$$\"3&om;z*ev:JFap$\"3#f.b0')[d V(F07$$\"3_LLL347TLFap$\"31JXX.J_CtF07$$\"3nLLLLY.KNFap$\"3\\dXAgA)HC( F07$$\"33++D\"o7Tv$Fap$\"3;lGk:\"f2;(F07$$\"3?LLL$Q*o]RFap$\"31)=>y(4# y4(F07$$\"3m++D\"=lj;%Fap$\"3+v'*oY?&z.(F07$$\"3S++vV&Rj _(o))pF07$$\"3CML$e9Ege%Fap$\"3Z(=]ZsRU%pF07$$\"3]LLeR\"3Gy%Fap$\"3$[= pRkZ)3pF07$$\"3emm;/T1&*\\Fap$\"35&>.?*RrvoF07$$\"3=nm\"zRQb@&Fap$\"39 yz>z%)3YoF07$$\"3:++v=>Y2aFap$\"3o@47#HRP#oF07$$\"3Znm;zXu9cFap$\"36:, w7yr-oF07$$\"34+++]y))GeFap$\"3Mi]FO1&Ry'F07$$\"3H++]i_QQgFap$\"3u<$QV N)4onF07$$\"3b++D\"y%3TiFap$\"3#=l$e2I![v'F07$$\"3+++]P![hY'Fap$\"3=%fr'F07$$\"3;,++D\\'QH(Fap$ \"3+N7nj<:4nF07$$\"3%HL$e9S8&\\(Fap$\"3PE`Gr&>Oq'F07$$\"3s++D1#=bq(Fap $\"3CpT2A`g)p'F07$$\"3\"HLL$3s?6zFap$\"3ojf@c9O%p'F07$$\"3a***\\7`Wl7) Fap$\"39#R2!G9_!p'F07$$\"3enmmm*RRL)Fap$\"3K3f(=EFto'F07$$\"3%zmmTvJga )Fap$\"3cOO,TE]%o'F07$$\"3]MLe9tOc()Fap$\"3u'p9W%H3#o'F07$$\"31,++]Qk \\*)Fap$\"3!*)GTo+],o'F07$$\"3![LL3dg6<*Fap$\"3y^KeK3BymF07$$\"3%ymmmw (Gp$*Fap$\"3wNu\"[$puwmF07$$\"3C++D\"oK0e*Fap$\"3%*Hp(p\"RPvmF07$$\"35 ,+v=5s#y*Fap$\"3g;OCz_BumF07$$FenF*$\"3gKLL$3xJn'F0-%&COLORG6&FR$F3!\" \"Fh`lFh`l-%*LINESTYLEG6#F--F$6%7W7$FSFS7$Fcp$\"3O%yn?A(o][!#>7$Fhp$\" 3S,1CYpV[$*Fdal7$F]q$\"3g?+W4)z%*H\"F07$Fbq$\"3/L\"=O?>4k\"F07$Fgq$\"3 G\">@aeB!**>F07$F\\r$\"3V)4WQh6;L#F07$Far$\"38qy#[stCk#F07$Ffr$\"3*)QA #*\\U/JHF07$F[s$\"3ggOKJAJXMF07$F`s$\"3Y'>h!H#Q'eQF07$Fes$\"3'yAr<.:3B %F07$Fjs$\"3U'f_X&R\"Rc%F07$F_t$\"3%*>\"*G')eg][F07$Fdt$\"3Ex%39_wZ5&F 07$Fit$\"3e)H[_=9!*H&F07$F^u$\"3m)RZATJ*)[&F07$Fcu$\"3=8u$)H%)4`cF07$F hu$\"37BmWuOh*y&F07$F]v$\"3m'HyFZ%e(*eF07$Fbv$\"3`,)y)H-\")3gF07$Fgv$ \"35v(3J2^.4'F07$F\\w$\"3Wn/pwX'F07$F_y$\"3Ya`8a[C(['F07$Fdy$\"3#4T77/%f4lF07$Fiy $\"3[\"Gk'F07$Fj]l$\"3GCuXrg+YmF07$F_^l$\"3/'p>BpI)[mF07$Fd^l$\"3'ej=*) Q]7l'F07$Fi^l$\"3qV?\\EL=`mF07$F^_l$\"3!33]2]-^l'F07$Fc_l$\"3%o*e^)R'e cmF07$Fh_l$\"3m-kN;%fzl'F07$F]`l$\"3+;(*3a!)4fmF07$Fb`l$\"3++++]i:gmF0 Fe`lFj`l-F$6%7S7$FS$\"3ImmmmmmmmF07$FhpFb[m7$FbqFb[m7$F\\rFb[m7$FfrFb[ m7$F[sFb[m7$F`sFb[m7$FesFb[m7$FjsFb[m7$F_tFb[m7$FdtFb[m7$FitFb[m7$F^uF b[m7$FcuFb[m7$FhuFb[m7$F]vFb[m7$FbvFb[m7$FgvFb[m7$F\\wFb[m7$FawFb[m7$F fwFb[m7$F[xFb[m7$F`xFb[m7$FexFb[m7$FjxFb[m7$F_yFb[m7$FdyFb[m7$FiyFb[m7 $F^zFb[m7$FczFb[m7$FhzFb[m7$F][lFb[m7$Fb[lFb[m7$Fg[lFb[m7$F\\\\lFb[m7$ Fa\\lFb[m7$Ff\\lFb[m7$F[]lFb[m7$F`]lFb[m7$Fe]lFb[m7$Fj]lFb[m7$F_^lFb[m 7$Fd^lFb[m7$Fi^lFb[m7$F^_lFb[m7$Fc_lFb[m7$Fh_lFb[m7$F]`lFb[m7$Fb`lFb[m -FP6&FR$\")!\\DP\"FVFf^m$\")viobFV-F[al6#F3-%+AXESLABELSG6%%\"xG%\"yG- %%FONTG6#%(DEFAULTG-%%VIEWG6$;FSFK;$!\"*!\"#$FjoFi`l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 135 "We can use the formula for the sum o f an infinite geometric series to express an infinite repeating decima l as a ratio of two integers." }}{PARA 0 "" 0 "" {TEXT -1 75 "For exam ple, consider the infinite repeating decimal: 0.123123123123 . . . " } }{PARA 0 "" 0 "" {TEXT -1 26 "This number has the form: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "123/(10^3)+123/(10^6)+123/(10^9 )+123/(10^12)+` . . . `;" "6#,,*&\"$B\"\"\"\"*$\"#5\"\"$!\"\"F&*&F%F&* $F(\"\"'F*F&*&F%F&*$F(\"\"*F*F&*&F%F&*$F(\"#7F*F&%(~.~.~.~GF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "This is an infinite geometric series with first term " }{XPPEDIT 18 0 "a = 123/(10^3);" "6#/%\"aG*& \"$B\"\"\"\"*$\"#5\"\"$!\"\"" }{TEXT -1 18 " and common ratio " } {XPPEDIT 18 0 "r=1/10^3" "6#/%\"rG*&\"\"\"F&*$\"#5\"\"$!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "The sum of this infinite geo metric series is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a/(1-r) = ``(123/(10^3))/(1-1/(10^3));" "6#/*&%\"aG\"\"\",&F&F&%\"rG! \"\"F)*&-%!G6#*&\"$B\"F&*$\"#5\"\"$F)F&,&F&F&*&F&F&*$F1F2F)F)F)" } {XPPEDIT 18 0 "``=123/(10^3-1)" "6#/%!G*&\"$B\"\"\"\",&*$\"#5\"\"$F'F' !\"\"F," }{XPPEDIT 18 0 "``=123/999" "6#/%!G*&\"$B\"\"\"\"\"$***!\"\" " }{XPPEDIT 18 0 "``=41/333" "6#/%!G*&\"#T\"\"\"\"$L$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 65 "As a (partial) check, a calculato r working with ten digits gives " }{XPPEDIT 18 0 "41/333" "6#*&\"#T\" \"\"\"$L$!\"\"" }{TEXT -1 1 " " }{TEXT 299 1 "~" }{TEXT -1 15 " 0.1231 231231. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "The distance travelled by a bouncing ball " }}{PARA 0 "" 0 "" {TEXT 310 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 58 "When a small hard rubber ball is dropped from a height of " }{TEXT 312 1 "h" }{TEXT -1 62 " metres onto a hard, flat \+ surface, it rebounds to a height of " }{XPPEDIT 18 0 "4/5;" "6#*&\"\"% \"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }{TEXT 313 1 "h" }{TEXT -1 9 " metre s. " }}{PARA 0 "" 0 "" {TEXT -1 55 "The ball is dropped initially from a height of 120 cm. " }}{PARA 0 "" 0 "" {TEXT -1 63 "(a) Find the hei ght to which the ball rises after two bounces. " }}{PARA 0 "" 0 "" {TEXT -1 100 "(b) Find the total distance that the ball has travelled \+ when it hits the surface for the 10th time. " }}{PARA 0 "" 0 "" {TEXT -1 51 "(c) Find the total distance travelled by the ball. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 311 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 82 "The following picture illustr ates the motion of the ball as a graph of the height " }{TEXT 314 1 "s " }{TEXT -1 51 " of the ball above the surface as time progresses. 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The ball is to be thought of as hav ing been given an initial horizontal velocity, so that it moves horizo ntally as well as up and down." }}{PARA 0 "" 0 "" {TEXT -1 207 "To obt ain a reasonably smooth animation a large number of frames are require d, and this consumes a lot of memory. It is probably not a good idea t o try and save the worksheet with the animation constructed. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 996 "h1 := 120: # initial height \nr := 0.8: # ratio of heights of bo unces\nnn := 10: # number of bounces\nfrms := 250: # number of frames \+ for animation\n\ng := 9.8: v1 := sqrt(2*h1*g): t0 := 0: t1 := v1/g:\nf or i from 2 to nn do\n v||i := v||(i-1)*sqrt(r);\n t||i := t||(i-1 )+2*v||i/g;\n h||i := (v||i)^2/(2*g):\nend do:\nH := x -> piecewise( x<0,0,1):\nv1*(t+t1)-g*(t+t1)^2/2+H(t-t1)*(-v1*(t+t1)+g*(t+t1)^2/2+v2* (t-t1)-g*(t-t1)^2/2)+\n add(H(t-t||i)*(-v||i*(t-t||(i-1))+g*(t-t||(i-1 ))^2/2+v||(i+1)*(t-t||i)-g*(t-t||i)^2/2),\n i=2..nn):\nB := unapply( %,t):\np1 := plot([[t,v1*(t+t1)-g*(t+t1)^2/2,t=0..t1],\n seq([t,v||(i+ 1)*(t-t||i)-g*(t-t||i)^2/2,t=t||i..t||(i+1)],i=1..nn-1)],\n color=CO LOR(RGB,.3,.3,.3)):\ntt := (t||nn)/frms:\npp := NULL:\nfor j from 0 to frms do\n p2 := plot([[[j*tt,B(tt*j)]]$4],style=point,symbol=[circl e,cross,diamond,circle],\n symbolsize=[10$3,12],color=[red$3,black] ):\n pp := pp,plots[display]([p1,p2]):\nend do:\nplots[display]([pp] ,insequence=true,xtickmarks=0,labels=[``,`s`]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 67 " Text Page 731, Ex 10.1 # 4-56, Page 750, Ex 10.3 #31,32,39 - \+ 42. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 29 "Find the sum of each series. " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "Sum(i^2,i=1..5)" "6#-%$SumG6$*$%\"iG\"\"#/ F';\"\"\"\"\"&" }{TEXT -1 10 " (b) " }{XPPEDIT 18 0 "Sum(n!,n=1.. 5)" "6#-%$SumG6$-%*factorialG6#%\"nG/F);\"\"\"\"\"&" }{TEXT -1 8 " \+ (c) " }{XPPEDIT 18 0 "Sum(1/i^2,i=1..5)" "6#-%$SumG6$*&\"\"\"F'*$%\"iG \"\"#!\"\"/F);F'\"\"&" }{TEXT -1 8 " (d) " }{XPPEDIT 18 0 "Sum(1/n! ,n=1..5)" "6#-%$SumG6$*&\"\"\"F'-%*factorialG6#%\"nG!\"\"/F+;F'\"\"&" }{TEXT -1 8 " (e) " }{XPPEDIT 18 0 "Sum((-1/3)^(k-1),k=1..5)" "6#-% $SumG6$),$*&\"\"\"F)\"\"$!\"\"F+,&%\"kGF)F)F+/F-;F)\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 21 "(a) 55 (b) 153 (c) " } {XPPEDIT 18 0 "5269/3600" "6#*&\"%p_\"\"\"\"%+O!\"\"" }{TEXT -1 6 " ( d) " }{XPPEDIT 18 0 "103/60" "6#*&\"$.\"\"\"\"\"#g!\"\"" }{TEXT -1 6 " (e) " }{XPPEDIT 18 0 "61/81" "6#*&\"#h\"\"\"\"#\")!\"\"" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "____________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 40 "Find the sum of each arithmetic series. " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "1+2+3+4+5+` . . . `+77" "6#,0\"\"\"F$\"\"#F$ \"\"$F$\"\"%F$\"\"&F$%(~.~.~.~GF$\"#xF$" }{TEXT -1 8 " (b) " } {XPPEDIT 18 0 "1+3+5+7+9+` . . .`+399;" "6#,0\"\"\"F$\"\"$F$\"\"&F$\" \"(F$\"\"*F$%'~.~.~.GF$\"$*RF$" }{TEXT -1 6 " (c) " }{XPPEDIT 18 0 "S um(48-i/3,i=0..60)" "6#-%$SumG6$,&\"#[\"\"\"*&%\"iGF(\"\"$!\"\"F,/F*; \"\"!\"#g" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " } {XPPEDIT 18 0 "1/2+3/4+1+5/4+` . . . `+19/4+5" "6#,0*&\"\"\"F%\"\"#!\" \"F%*&\"\"$F%\"\"%F'F%F%F%*&\"\"&F%F*F'F%%(~.~.~.~GF%*&\"#>F%F*F'F%F,F %" }{TEXT -1 7 " (e) " }{XPPEDIT 18 0 "1+4/3+5/3+2+` . . . `+28/3;" "6#,.\"\"\"F$*&\"\"%F$\"\"$!\"\"F$*&\"\"&F$F'F(F$\"\"#F$%(~.~.~.~GF$*& \"#GF$F'F(F$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 35 "(a) 3003 (b) 40000 (c) 2318 (d) " }{XPPEDIT 18 0 "209/4" "6#*&\"$4#\" \"\"\"\"%!\"\"" }{TEXT -1 6 " (e) " }{XPPEDIT 18 0 "403/3" "6#*&\"$.% \"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "____________________ _______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 83 "An Engl ish teacher with a minor in mathematics told her students that if they read " }{XPPEDIT 18 0 "2*n+1" "6#,&*&\"\"#\"\"\"%\"nGF&F&F&F&" } {TEXT -1 30 " pages of a long novel on the " }{TEXT 301 1 "n" }{TEXT -1 151 " th day of May, starting on the 1st of May, then they would fi nish the novel exactly on the last day of the month. How many pages do es the novel have? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 "1023 " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "____________________ _______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 85 "(a) Fin d the number of terms of a geometric sequence with first term 7, commo n ratio " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 16 " , and last term " }{XPPEDIT 18 0 "7/1024" "6#*&\"\"(\"\"\"\"%C5!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 66 "(b) Find the first ter m of a geometric sequence with common ratio " }{XPPEDIT 18 0 "1/3" "6# *&\"\"\"F$\"\"$!\"\"" }{TEXT -1 16 " and sixth term " }{XPPEDIT 18 0 " 1/243" "6#*&\"\"\"F$\"$V#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "(c) Find a formula for the " }{TEXT 302 1 "n" }{TEXT -1 69 " th term of a geometric sequence with second term -20 and fifth te rm " }{XPPEDIT 18 0 "1/50" "6#*&\"\"\"F$\"#]!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "An s " }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) 11 (b) 3 (c) " }{XPPEDIT 18 0 "-1/10" "6#,$*&\"\"\"F%\"#5!\"\"F'" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 " ___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 48 "Find the sum of each \+ (finite) geometric series. " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " } {XPPEDIT 18 0 "2+10+50+250+1250+6250+31250" "6#,0\"\"#\"\"\"\"#5F%\"#] F%\"$]#F%\"%]7F%\"%]iF%\"&]7$F%" }{TEXT -1 9 " (b) " }{XPPEDIT 18 0 "6+2+2/3+2/9+2/27+2/81+2/243+2/729" "6#,2\"\"'\"\"\"\"\"#F%*&F&F%\" \"$!\"\"F%*&F&F%\"\"*F)F%*&F&F%\"#FF)F%*&F&F%\"#\")F)F%*&F&F%\"$V#F)F% *&F&F%\"$H(F)F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(c) " }{XPPEDIT 18 0 "Sum(3*(1/2)^(i-1),i=1..8)" "6#-%$SumG6$*&\"\"$\"\"\")* &F(F(\"\"#!\"\",&%\"iGF(F(F,F(/F.;F(\"\")" }{TEXT -1 13 " (d) \+ " }{XPPEDIT 18 0 "Sum(5*(1/10)^(i-1),i=1..10)" "6#-%$SumG6$*&\"\"&\"\" \")*&F(F(\"#5!\"\",&%\"iGF(F(F,F(/F.;F(F+" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }} {PARA 0 "" 0 "" {TEXT -1 17 "(a) 39062 (b) " }{XPPEDIT 18 0 "6560/7 29" "6#*&\"%gl\"\"\"\"$H(!\"\"" }{TEXT -1 7 " (c) " }{XPPEDIT 18 0 " 765/128" "6#*&\"$l(\"\"\"\"$G\"!\"\"" }{TEXT -1 7 " (d) " }{XPPEDIT 18 0 "1111111111/200000000=5" "6#/*&\"+66666\"\"\"\"*++++#!\"\"\"\"&" }{TEXT -1 11 ".555555555 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6" }} {PARA 0 "" 0 "" {TEXT -1 66 "Where possible, find the sum of each (inf inite) geometric series. " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " } {XPPEDIT 18 0 "3+1+1/3+1/9+1/27+` . . . `;" "6#,.\"\"$\"\"\"F%F%*&F%F% F$!\"\"F%*&F%F%\"\"*F'F%*&F%F%\"#FF'F%%(~.~.~.~GF%" }{TEXT -1 12 " \+ (b) " }{XPPEDIT 18 0 "1-1/3+1/9-1/27+1/81-` . . . `;" "6#,.\"\"\"F $*&F$F$\"\"$!\"\"F'*&F$F$\"\"*F'F$*&F$F$\"#FF'F'*&F$F$\"#\")F'F$%(~.~. ~.~GF'" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "(c) " }{XPPEDIT 18 0 "1/3+2/9+4/27+8/81+` . . . `" "6#,,*&\"\"\"F%\"\"$!\"\"F%*&\"\"#F%\"\"*F'F%*&\"\"%F%\"#FF'F%*&\"\")F %\"#\")F'F%%(~.~.~.~GF%" }{TEXT -1 9 " (d) " }{XPPEDIT 18 0 "7/10+ 7/100+7/1000+7/10000+7/100000+` . . . `" "6#,.*&\"\"(\"\"\"\"#5!\"\"F& *&F%F&\"$+\"F(F&*&F%F&\"%+5F(F&*&F%F&\"&++\"F(F&*&F%F&\"'++5F(F&%(~.~. ~.~GF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 5 "(e) " }{XPPEDIT 18 0 "Sum(3*(3/7)^(i-1),i = 1 .. i nfinity);" "6#-%$SumG6$*&\"\"$\"\"\")*&F'F(\"\"(!\"\",&%\"iGF(F(F,F(/F .;F(%)infinityG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "9/2" "6#*&\"\"*\"\"\"\"\"#!\"\"" }{TEXT -1 8 " (b) " }{XPPEDIT 18 0 "3/4;" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 14 " (c) 1 (d) " }{XPPEDIT 18 0 "7/9" "6#*&\"\"(\"\"\"\"\"*!\"\" " }{TEXT -1 6 " (e) " }{XPPEDIT 18 0 "21/4" "6#*&\"#@\"\"\"\"\"%!\"\" " }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }} {PARA 0 "" 0 "" {TEXT -1 140 "Use the formula for the sum of an (infin ite) geometric series to express each infinite repeating decimal as a \+ fraction in its lowest terms. " }}{PARA 0 "" 0 "" {TEXT -1 80 "(a) 0.2 1212121 . . . (b) 0.252252252252 . . . (c) 0.5544554455445544 . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "7/33; " "6#*&\"\"(\"\"\"\"#L!\"\"" }{TEXT -1 8 " (b) " }{XPPEDIT 18 0 "28 /11;" "6#*&\"#G\"\"\"\"#6!\"\"" }{TEXT -1 7 " (c) " }{XPPEDIT 18 0 " 56/101" "6#*&\"#c\"\"\"\"$,\"!\"\"" }{TEXT -1 3 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "______ _____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q8" }}{PARA 0 "" 0 "" {TEXT -1 185 "(a) If you deposit $ 1200 at the beginning of each year for 30 years into an account paying 7% interest compounded annually, how much is in the account at the en d of the 30 year period? " }}{PARA 0 "" 0 "" {TEXT -1 260 "(b) If $100 dollars is deposited at the end of each month for 30 years into an ac count earning 7% effective annual interest rate compounded monthly, ho w much is in the account at the end of the 30 year period. (Exclude a \+ deposit at the end of the last month.) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "(a) $121287.65 (b) $121 897.10 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "(1200*(1.07)*((1.07)^30-1)/0.07);\nevalf[8](%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'\\wG@\"!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")l(G@\"!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "100*(1+7/1200)*((1+7/1200)^3 59-1)/(7/1200):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")5 (*=7!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "__________________ _________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q9 " }}{PARA 0 "" 0 "" {TEXT -1 109 "Repeat the bouncing ball example if \+ the ball is dropped from a height of 120 cm. and rebounds to a height \+ of " }{XPPEDIT 18 0 "3/5*h" "6#*(\"\"$\"\"\"\"\"&!\"\"%\"hGF%" }{TEXT -1 28 " when dropped from a height " }{TEXT 317 1 "h" }{TEXT -1 2 ". \+ " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "216/5=43. 2" "6#/*&\"$;#\"\"\"\"\"&!\"\"-%&FloatG6$\"$K%F(" }{TEXT -1 11 " cm. \+ (b) " }{XPPEDIT 18 0 "120+360*(1-(3/5)^9)" "6#,&\"$?\"\"\"\"*&\"$g$F% ,&F%F%*$*&\"\"$F%\"\"&!\"\"\"\"*F-F%F%" }{TEXT -1 1 " " }{TEXT 318 1 " ~" }{TEXT -1 25 " 476.37 cm. (c) 480 cm. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "(3/5)^2*120;\neval f(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$;#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++?V!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "120+360*(1-(3/5)^9);\nevalf( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"*CG3'=\"'D1R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%H?Pw%!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "120+240*(3/5)/(1-3/5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$![" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 27 "___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 27 "____________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 18 "Code for pi ctures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 36 "Code for sum of odd numbers picture " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 589 "t1:=plots[textplot]([seq([i,1,2*i-1],i=1..4),seq([i-90,1,2*i-1],i =97..100),\nseq([i+.5,1,`+`],i=1..4),seq([i-90.5,1,`+`],i=97..100),\ns eq([i,0,201-2*i],i=1..4),seq([i-90,0,201-2*i],i=97..100),\nseq([i+.5,0 ,`+`],i=1..4),seq([i-90.5,0,`+`],i=97..100)]):\nt2:=plots[textplot]([[ 5.5,1,`. . .`],[5.5,0,`. . .`]],font=[TIMES,BOLD,10]):\nt3:=plots[ textplot]([[0.4,1,`S =`],[0.33,0,`S =`]],font=[TIMES,ITALIC,12]):\np 1:=plot([seq([[i-.23,1.3],[i+.23,1.3],[i+.23,-.3],[i-.23,-.3],[i-.23,1 .3]],\n i=[1,2,3,4,7,8,9,10])],color=COLOR(RGB,.8,0,0),thickness=2): \+ \nplots[display]([t1,t2,t3,p1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 754 "t1:=plots[textplo t]([seq([i,1,2*i-1],i=1..4),seq([i-90,1,2*i-1],i=97..100),\nseq([i+.5, 1,`+`],i=1..4),seq([i-90.5,1,`+`],i=97..100),\nseq([i,0,201-2*i],i=1.. 4),seq([i-90,0,201-2*i],i=97..100),\nseq([i+.5,0,`+`],i=1..4),seq([i-9 0.5,0,`+`],i=97..100),\nseq([i,-1,200],i=1..4),seq([i-90,-1,200],i=97. .100),\nseq([i+.5,-1,`+`],i=1..4),seq([i-90.5,-1,`+`],i=97..100),[0,-1 ,2]]):\nt2:=plots[textplot]([[5.5,1,`. . .`],[5.5,0,`. . .`],\n \+ [5.5,-1,`. . .`]],font=[TIMES,BOLD,10]):\nt3:=plots[textplot]([[0.4 ,1,`S =`],[0.33,0,`S =`],\n [0.33,-1,`S =`]],font=[TIMES,ITALIC,1 2]):\np1:=plot([seq([[i-.23,1.3],[i+.23,1.3],[i+.23,-.3],[i-.23,-.3],[ i-.23,1.3]],\n i=[1,2,3,4,7,8,9,10])],color=COLOR(RGB,.8,0,0),thickne ss=2): \nplots[display]([t1,t2,t3,p1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 31 "Code for bouncing ball picture " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 464 "h1 := 2:\nr := 0.8: g := 9.8:\nv1 := sqrt(2*h1*g): t1 := 2*v1/g: \nfor i from 2 to 10 do\n v||i := v||(i-1)*sqrt(r);\n t||i := t||( i-1)+2*v||i/g;\n h||i := (v||i)^2/(2*g):\nend do:\np1 := plot([[t,v1 *t-g*t^2/2,t=0..t1],[t,v2*(t-t1)-g*(t-t1)^2/2,t=t1..t2],\n [t,v3*(t-t 2)-g*(t-t2)^2/2,t=t2..t3]],color=red):\np2 := plot([[[t1/2,0],[t1/2,h1 ]],[[(t1+t2)/2,0],[(t1+t2)/2,h2]],\n[[(t2+t3)/2,0],[(t2+t3)/2,h3]]],co lor=navy,linestyle=2):\nplots[display]([p1,p2],xtickmarks=0);\n" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 460 "h1 := 5:\nr := 0.8: \ng := 9.8:\nnn := 20:\nv1 := sqrt(2*h1*g): t 0 := 0: t1 := 2*v1/g:\nfor i from 2 to nn do\n v||i := v||(i-1)*sqrt (r);\n t||i := t||(i-1)+2*v||i/g;\n h||i := (v||i)^2/(2*g):\nend d o:\np1 := plot([seq([t,v||(i+1)*(t-t||i)-g*(t-t||i)^2/2,t=t||i..t||(i+ 1)],i=0..nn-1)],\n color=red):\np2 := plot([seq([[(t||i+t||(i+1))/2,0 ],[(t||i+t||(i+1))/2,h||(i+1)]],i=0..nn-1)],\n color=navy,linestyle= 2):\nplots[display]([p1,p2],xtickmarks=8,labels=[`t`,`h`]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 509 "h 1 := 120:\nr := 0.8: \ng := 9.8:\nnn := 20:\nv1 := sqrt(2*h1*g): t0 := 0: t1 := v1/g:\nfor i from 2 to nn do\n v||i := v||(i-1)*sqrt(r);\n t||i := t||(i-1)+2*v||i/g;\n h||i := (v||i)^2/(2*g):\nend do:\np1 := plot([[t,v1*(t+t1)-g*(t+t1)^2/2,t=0..t1],\n seq([t,v||(i+1)*(t-t|| i)-g*(t-t||i)^2/2,t=t||i..t||(i+1)],i=1..nn-1)],color=red,\n thicknes s=2):\np2 := plot([seq([[(t||i+t||(i+1))/2,0],[(t||i+t||(i+1))/2,h||(i +1)]],i=1..nn-1)],\n color=navy,linestyle=2):\nplots[display]([p1,p2 ],xtickmarks=0,labels=[`t`,`s`]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }