{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 128 1 0 0 2 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "Courier" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 273 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 128 0 1 0 1 2 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 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 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 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 "Map le Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 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 "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "Tit le" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 257 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 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 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 "" 256 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 19 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 263 1 {CSTYLE "" -1 -1 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 " Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 259 "" 0 "" {TEXT 273 36 "Linear Recurrence Relat ions in Maple" }}{PARA 260 "" 0 "" {TEXT 270 46 "Date: Nov 7, 2001\nLa st Revision: Nov 15, 2001\n" }{TEXT 282 7 "Maple 6" }}{PARA 265 "" 0 " " {TEXT 274 16 "Bent E. Petersen" }}{PARA 263 "" 0 "" {TEXT 275 17 "be nt@alum.mit.edu" }}{PARA 264 "" 0 "" {TEXT 276 22 "petersen@math.orst. edu" }}{PARA 0 "" 0 "" {TEXT 277 0 "" }}{PARA 0 "" 0 "" {TEXT 278 32 " Course: Mth 355 (a.k.a. Mth 399)" }}{PARA 0 "" 0 "" {TEXT 279 15 "Term : Fall 2001" }}{PARA 0 "" 0 "" {TEXT 280 11 "File name: " }{TEXT 272 18 "355f2001_recur.mws" }}{PARA 0 "" 0 "" {TEXT 281 50 "Assignment 6 ( problems at end) - due Nov 14, 2001." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 227 "This worksheet contains a few comments on some of Maple's recurrence relations support. It composed fairly q uickly and is unlikely to be free of errors. If you find a serious err or be sure to mention it in your solution report." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "L et's load the plots package - just in case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots) :" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has b een redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 " " 0 "" {TEXT -1 9 "Example 1" }{TEXT 261 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "The Maple function rsolve() is used to solve linear recurrence relations. Here is an example of an o rder 2 recurrence relation with initial conditions." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqn1:=a(n )=a(n-1)+a(n-2); # Fibonacci" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eq n1G/-%\"aG6#%\"nG,&-F'6#,&F)\"\"\"F.!\"\"F.-F'6#,&F)F.\"\"#F/F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "init1:=a(0)=0,a(1)=1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init1G6$/-%\"aG6#\"\"!F*/-F(6#\"\" \"F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "To solve we throw all the equations into one set and also speci fy for which variable to solve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "soln1:=rsolve(\{eqn1,init1\} ,a); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&soln1G,&*&*&,&*$-%%sqrtG6# \"\"&\"\"\"#F.F-F.!\"\"F.),$*&F.F.,&F.F.F)F0F0!\"#%\"nGF.F.F4F0F.*&*&, &F)#F0F-F.F0F.),$*&F.F.,&F.F.F)F.F0F5F6F.F.F>F0F." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 157 "That's all there \+ is to it! This expression is not in the usual Binet form, so you may w ant to try to get Maple to cast it in a more familiar form. Good luck! " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 "We c an also use rsolve() to find the generating function for the solutio n. Note the single quotes surrounding genfunc in the following comman d." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "gen1:=rsolve(\{eqn1,init1\},a,'genfunc'(z));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gen1G,$*&%\"zG\"\"\",(!\"\"F(F'F(*$ )F'\"\"#F(F(F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 60 "We can even define a procedure which will compute \+ the a(n)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "fun1:=rsolve(\{eqn1,init1\},a,'makeproc'):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Th e fun1() procedure can be use to plot the sequence a(n)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sq1 :=seq([n,fun1(n)],n=0..16);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sq1G 637$\"\"!F'7$\"\"\"F)7$\"\"#F)7$\"\"$F+7$\"\"%F-7$\"\"&F17$\"\"'\"\")7 $\"\"(\"#87$F4\"#@7$\"\"*\"#M7$\"#5\"#b7$\"#6\"#*)7$\"#7\"$W\"7$F7\"$L #7$\"#9\"$x$7$\"#:\"$5'7$\"#;\"$()*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "PLOT(POINTS(sq1),SYMBOL(BOX,16),TEXT([5,1000],\"fun1 \+ - Fibonacci sequence\"));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'POINTSG637$\"\"!F'7$\"\"\"F)7$\"\"#F)7$\"\"$F+7$\"\" %F-7$\"\"&F17$\"\"'\"\")7$\"\"(\"#87$F4\"#@7$\"\"*\"#M7$\"#5\"#b7$\"#6 \"#*)7$\"#7\"$W\"7$F7\"$L#7$\"#9\"$x$7$\"#:\"$5'7$\"#;\"$()*-%'SYMBOLG 6$%$BOXGFO-%%TEXTG6$7$F1\"%+5Q:fun1~-~Fibonacci~sequence6\"" 1 2 3 1 16 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }} }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " We could plot a curve instead" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "PLOT(CURVES([sq1]),TEXT([5,9 00],\"fun1 - Fibonacci sequence\"));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6$-%'CURVESG6#737$\"\"!F(7$\"\"\"F*7$\"\"#F*7 $\"\"$F,7$\"\"%F.7$\"\"&F27$\"\"'\"\")7$\"\"(\"#87$F5\"#@7$\"\"*\"#M7$ \"#5\"#b7$\"#6\"#*)7$\"#7\"$W\"7$F8\"$L#7$\"#9\"$x$7$\"#:\"$5'7$\"#;\" $()*-%%TEXTG6$7$F2\"$+*Q:fun1~-~Fibonacci~sequence6\"" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "He re's one (kludgy) way to get the characteristic polynomial and the cha racteristic root" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "subs(a(n)=x^n,a(n-1)=x^(n-1),a(n-2)=x^(n-2),e qn1)/x^(n-2): cpoly1:=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%'cpoly1G/*$)%\"xG\"\"#\"\"\",&F(F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "croot1:=solve(cpoly1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'croot1G6$,&#\"\"\"\"\"#F(*&F'F(-%%sqrtG6#\"\"&F(F(,& F'F(*&#F(F)F(*$F+F(F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 201 "No te it is not too difficult to write a procedure to compute the order o f the recurrence equation and then to compute the characteristic polyn omial. I will do this in the next worksheet (Assignment 7)." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 9 "Ex ample 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Here's an inhomogeneous e quation related to the Fibonacci equation" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "eqn2:=a(n)=a(n-1)+a(n-2)-n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%\"aG6#%\"nG,(-F'6#,&F)\"\"\"F.!\"\"F.-F'6#,& F)F.\"\"#F/F.F)F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "init2: =a(0)=0,a(1)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init2G6$/-%\"aG6 #\"\"!F*/-F(6#\"\"\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "s oln2:=rsolve(\{eqn2,init2\},a); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% &soln2G,.*&*&,&*$-%%sqrtG6#\"\"&\"\"\"#F.F-F.!\"\"F.),$*&F.F.,&F.F.F)F 0F0!\"#%\"nGF.F.F4F0F.*&*&,&F)#F0F-F.F0F.),$*&F.F.,&F.F.F)F.F0F5F6F.F. F>F0F.F6F.\"\"$F.*&*&F>F.F1F.F.F4F0F.*&*&F4F.F;F.F.F>F0F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "gen2:=rsolve(\{eqn2,init2\},a,'genf unc'(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gen2G,$*&,&*&,&*&\"\" \"F+,&F+F+%\"zG!\"\"F.!\"#*&F-F+*$)F,\"\"#F+F.F.F+)F-F3F+F+F-F+F+,(F.F +F-F+*$F4F+F+F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "fun2:= rsolve(\{eqn2,init2\},a,'makeproc'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sq2:=seq([n,fun2(n)],n=0..20):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 68 "PLOT(POINTS(evalf(sq2)),SYMBOL(DIAMOND,16),TEX T([5,-20000],\"fun2\"));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'POINTSG677$$\"\"!F(F'7$$\"\"\"F($\"+-+++5!\"*7$$\"\" #F($!*$********F.7$$\"\"$F($!+()******HF.7$$\"\"%F($!*(******z!\")7$$ \"\"&F($!+%******f\"F>7$$\"\"'F($!+!*******HF>7$$\"\"(F($!+!)*****H&F> 7$$\"\")F($!*'*****4*!\"(7$$\"\"*F($!+%*****H:FS7$$\"#5F($!+))****RDFS 7$$\"#6F($!+!)****zTFS7$$\"#7F($!+j****RoFS7$$\"#8F($!+$****\\6\"!\"'7 $$\"#9F($!+!****H\"=Fgo7$$\"#:F($!+\")***H%HFgo7$$\"#;F($!+o***>x%Fgo7 $$\"#t(Fgo7$$\"#=F($!+!***>_7!\"&7$$\"#>F($!+$)**HF?Faq7$$ \"#?F($!+r**\\\"G$Faq-%'SYMBOLG6$%(DIAMONDGFdp-%%TEXTG6$7$FA!&++#Q%fun 26\"" 1 2 2 1 16 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 263 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn3:=a(n)=a(n-1)-a(n-2)/2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%\"aG6#%\"nG,&-F'6#,&F)\"\" \"F.!\"\"F.*&#F.\"\"#F.-F'6#,&F)F.F2F/F.F/" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "init3:=a(0)=1,a(1)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init3G6$/-%\"aG6#\"\"!\"\"\"/-F(6#F+F+" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "soln3:=rsolve(\{eqn3,init3\},a); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&soln3G,&*&^$#\"\"\"\"\"#F(F))^$F(#!\"\"F*%\" nGF)F)*&F,F))F'F/F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ge n3:=rsolve(\{eqn3,init3\},a,'genfunc'(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gen3G,$*&\"\"\"F',(\"\"#F'*&F)F'%\"zGF'!\"\"*$)F+F)F 'F'F,F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "fun3:=rsolve(\{e qn3,init3\},a,'makeproc'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sq3:=seq([n,fun3(n)],n=0..20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "PLOT(POINTS(evalf(sq3)),SYMBOL(DIAMOND,16),TEXT([20,- 1],\"fun3\"));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 " 6%-%'POINTSG677$$\"\"!F($\"\"\"F(7$F)F)7$$\"\"#F($\"+++++]!#57$$\"\"$F (F'7$$\"\"%F($!+++++DF17$$\"\"&F(F87$$\"\"'F($!++++]7F17$$\"\"(F(F'7$$ \"\")F($\"++++]i!#67$$\"\"*F(FH7$$\"#5F($\"++++DJFJ7$$\"#6F(F'7$$\"#7F ($!+++]i:FJ7$$\"#8F(FY7$$\"#9F($!+++]7y!#77$$\"#:F(F'7$$\"#;F($\"+++D1 RF]o7$$\"#F]o7$$\"#>F(F'7$$\"#?F($!++]il(*!#8- %'SYMBOLG6$%(DIAMONDGFco-%%TEXTG6$7$Fcp!\"\"Q%fun36\"" 1 2 2 1 16 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "PLOT(CURVES([evalf(sq3)]),SY MBOL(DIAMOND,16),TEXT([4,0.9],\"fun3\"));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6#777$$\"\"!F)$\"\"\"F) 7$F*F*7$$\"\"#F)$\"+++++]!#57$$\"\"$F)F(7$$\"\"%F)$!+++++DF27$$\"\"&F) F97$$\"\"'F)$!++++]7F27$$\"\"(F)F(7$$\"\")F)$\"++++]i!#67$$\"\"*F)FI7$ $\"#5F)$\"++++DJFK7$$\"#6F)F(7$$\"#7F)$!+++]i:FK7$$\"#8F)FZ7$$\"#9F)$! +++]7y!#77$$\"#:F)F(7$$\"#;F)$\"+++D1RF^o7$$\"#F^o7$$\"#>F)F(7$$\"#?F)$!++]il(*!#8-%'SYMBOLG6$%(DIAMONDGFdo-%%TEXTG6 $7$F8$FN!\"\"Q%fun36\"" 1 2 2 1 16 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "subs(a(n)=x^n,a(n-1)=x^(n-1),a(n-2)=x^(n-2),eqn3)/x^( n-2): cpoly3:=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cpoly 3G/*$)%\"xG\"\"#\"\"\",&F(F*#F*F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "croot3:=solve(cpoly3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'croot3G6$^$#\"\"\"\"\"#F'^$F'#!\"\"F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 9 "Example 4 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eqn4:=a(n)=-a(n-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%eqn4G/-%\"aG6#%\"nG,$-F'6#,&F)\"\"\"\"\"#!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "init4:=a(0)=2,a(1)=-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init4G6$/-%\"aG6#\"\"!\"\"#/-F(6#\"\"\"!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "soln4:=rsolve(\{eqn4,in it4\},a); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&soln4G,&*&^$\"\"\"#! \"\"\"\"#F()^#F*%\"nGF(F(*&^$F(#F(F+F()^#F(F.F(F(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 42 "gen4:=rsolve(\{eqn4,init4\},a,'genfunc'(z)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gen4G*&,&\"\"#\"\"\"%\"zG!\"\"F (,&F(F(*$)F)F'F(F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "fun 4:=rsolve(\{eqn4,init4\},a,'makeproc'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sq4:=seq([n,fun4(n)],n=0..20):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 63 "PLOT(POINTS(evalf(sq4)),SYMBOL(DIAMOND,20),TEX T([3,3],\"fun4\"));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'POINTSG677$$\"\"!F($\"\"#F(7$$\"\"\"F($!\"\"F(7$F)$! \"#F(7$$\"\"$F(F,7$$\"\"%F(F)7$$\"\"&F(F.7$$\"\"'F(F17$$\"\"(F(F,7$$\" \")F(F)7$$\"\"*F(F.7$$\"#5F(F17$$\"#6F(F,7$$\"#7F(F)7$$\"#8F(F.7$$\"#9 F(F17$$\"#:F(F,7$$\"#;F(F)7$$\"#F(F,7$$\"#?F(F) -%'SYMBOLG6$%(DIAMONDGFbo-%%TEXTG6$7$F5F5Q%fun46\"" 1 2 2 1 20 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "PLOT(CURVES([evalf(sq4)]),SYMBOL(DI AMOND,16),TEXT([2,1.8],\"fun3\"));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6#777$$\"\"!F)$\"\"#F)7$$\"\"\"F)$!\" \"F)7$F*$!\"#F)7$$\"\"$F)F-7$$\"\"%F)F*7$$\"\"&F)F/7$$\"\"'F)F27$$\"\" (F)F-7$$\"\")F)F*7$$\"\"*F)F/7$$\"#5F)F27$$\"#6F)F-7$$\"#7F)F*7$$\"#8F )F/7$$\"#9F)F27$$\"#:F)F-7$$\"#;F)F*7$$\"#F)F-7 $$\"#?F)F*-%'SYMBOLG6$%(DIAMONDGFgn-%%TEXTG6$7$F+$F]oF0Q%fun36\"" 1 2 2 1 16 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Here we have a periodic solution." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "subs(a(n)=x^n,a(n-1)=x^ (n-1),a(n-2)=x^(n-2),eqn4)/x^(n-2): cpoly4:=simplify(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'cpoly4G/*$)%\"xG\"\"#\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "croot4:=solve(cpoly4,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'croot4G6$^#\"\"\"^#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 9 "Example 5 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eqn5:=a(n)=3*a(n-1)+4*a(n-2)-12*a(n-3);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%eqn5G/-%\"aG6#%\"nG,(-F'6#,&F)\"\"\"F.!\"\"\" \"$*&\"\"%F.-F'6#,&F)F.\"\"#F/F.F.*&\"#7F.-F'6#,&F)F.F0F/F.F/" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "subs(seq(a(n-k)=x^(n-k),k=0. .3),eqn5)/x^(n-3): cpoly5:=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cpoly5G/*$)%\"xG\"\"$\"\"\",(*$)F(\"\"#F*F)*&\"\"%F*F(F*F*\"# 7!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "croot5:=solve(cpo ly5,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'croot5G6%!\"#\"\"#\"\"$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "init5:=a(0)=2,a(1)=5,a( 2)=13;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init5G6%/-%\"aG6#\"\"!\" \"#/-F(6#\"\"\"\"\"&/-F(6#F+\"#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "soln5:=rsolve(\{eqn5,init5\},a); " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&soln5G,&)\"\"$%\"nG\"\"\")\"\"#F(F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 9 "Example \+ 6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 439 "Let a(n) be the number of strings of 0's and 1's of length \+ n which do not contain the bit pattern 11. If a such a string star ts with 0 the remainder can be any string of length n-1 which does not contain 11. If on the other hand it starts with 1 then the se cond character must be 0 and after that we can have any string of le ngth n-2 which does not contain 11. Thus (note we just have a shift ed Fibonacci sequence) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn6:=a(n)=a(n-1)+a(n-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/-%\"aG6#%\"nG,&-F'6#,&F)\"\"\"F.!\" \"F.-F'6#,&F)F.\"\"#F/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "init6:=a(0)=1,a(1)=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init6G6$/ -%\"aG6#\"\"!\"\"\"/-F(6#F+\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "soln6:=rsolve(\{eqn6,init6\},a(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&soln6G,&*&*&,&*$-%%sqrtG6#\"\"&\"\"\"#!\"\"F-F.F0 F.),$*&F.F.,&F.F.F)F0F0!\"#%\"nGF.F.F4F0F.*&*&,&F)#F.F-F.F0F.),$*&F.F. ,&F.F.F)F.F0F5F6F.F.F>F0F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "A6:=unapply(round(soln6),n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #A6GR6#%\"nG6\"6$%)operatorG%&arrowGF(-%&roundG6#,&*&*&,&*$-%%sqrtG6# \"\"&\"\"\"#!\"\"F7F8F:F8),$*&F8F8,&F8F8F3F:F:!\"#9$F8F8F>F:F8*&*&,&F3 #F8F7F8F:F8),$*&F8F8,&F8F8F3F8F:F?F@F8F8FHF:F8F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "for k from 0 to 14 do printf(\"%5d\", A6( k)); od;" }}{PARA 6 "" 1 "" {TEXT -1 75 " 1 2 3 5 8 1 3 21 34 55 89 144 233 377 610 987" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 9 "Example 7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 313 "Consid er n nonparallel lines in the plane and suppose no three of them mee t in a point. The lines divide the plane into a(n) regions. If we al ready have n-1 lines and add one more line, it cuts each of the exis ting n-1 lines and so creates n new regions. (This statement may r equire some argument.) Thus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eqn7:=a(n)=a(n-1)+n;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn7G/-%\"aG6#%\"nG,&-F'6#,&F)\"\" \"F.!\"\"F.F)F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "init7:=a (0)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&init7G/-%\"aG6#\"\"!\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "soln7:=rsolve(\{eqn7, init7\},a(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&soln7G,&*&,&%\"nG \"\"\"F)F)F),&F(#F)\"\"#F)F)F)F)F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "for k from 0 to 20 do printf(\"%4d\", subs(n=k,soln7) ); od;" }}{PARA 6 "" 1 "" {TEXT -1 84 " 1 2 4 7 11 16 22 2 9 37 46 56 67 79 92 106 121 137 154 172 191 211" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "rsolve(\{eqn7,init7\},a,'genfunc'(z )): gen7:=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%gen7G,$*& ,(%\"zG!\"\"\"\"\"F**$)F(\"\"#F*F*F**$),&F)F*F(F*\"\"$F*F)F)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 9 "E xample 8" }{TEXT 267 0 "" }{TEXT 268 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 189 "Suppose A is a set of c ardinality n, for convenience say, A = \{1,2,...,n\}. Let S(n,m) \+ be the number of partitions of A into m subsets (so the Stirling n umber of the second kind)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "The partitions which do contain the set \{n\} \+ correspond to partitions of \{1,2,...,n-1\} into m-1 subsets. Ther e are S(n-1,m-1) such partitions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 238 "The partitions which do not contain \{ n\} correspond to partitions of \{1,2,...,n-1\} into m subsets an y one of which we can add n to in order to obtain a partition of \{ 1,2,...,n\}. There are m S(n-1,m) such partitions. Thus we have" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eqn8:=S(n,m)=S(n-1,m-1)+m*S(n-1,m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn8G/-%\"SG6$%\"nG%\"mG,&-F'6$,&F)\"\"\"F/!\"\",&F* F/F/F0F/*&F*F/-F'6$F.F*F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "init8:=S(n,1)=1,S(n,n)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&in it8G6$/-%\"SG6$%\"nG\"\"\"F+/-F(6$F*F*F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "Here we have an example \+ of a double recurrence relation. Maple does not seem to be able to han dle such recurrences.." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 9 "Exampl e 9" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 242 "Consider a string of length n+1. The number of ways we can i nsert n pairs of parantheses so as to form n \"products\" of pairs of characters in the string is Cat(n) (a Catalan number). Cat(n) sa tisfies a first order recurrence equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn9:=(n+1)*c(n)= (4*n-2)*c(n-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn9G/*&,&%\"nG \"\"\"F)F)F)-%\"cG6#F(F)*&,&F(\"\"%\"\"#!\"\"F)-F+6#,&F(F)F)F1F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "init9:=c(1)=1;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&init9G/-%\"cG6#\"\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "rsolve(\{eqn9,init9\},c(n)): Cat:=unapply(%,n );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$CatGR6#%\"nG6\"6$%)operatorG% &arrowGF(*&*&)\"\"%9$\"\"\"-%&GAMMAG6#,&F0F1#F1\"\"#F1F1F1*&-F36#,&F0F 1F7F1F1-%%sqrtG6#%#PiGF1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "It is difficult to directly sim plify this expression. However we can get Maple to verify that " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "'Cat(n)' = 1/(n+1)*binomial(2*n,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$CatG6#%\"nG*&-%)binomialG6$,$F'\"\"#F'\"\"\",&F'F.F .F.!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Here's the verification - actually an automated proof of \+ sorts!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "simplify(Cat(n)-binomial(2*n,n)/(n+1));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 269 8 "Problems" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Solve the following recursive equations. In each case find the gen erating function as well." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Problem 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a(n) = 2*a(n-1)-7; a(0)=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#%\"nG,&-F%6#,&F'\"\"\"F,!\"\"\"\"#\"\"(F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#\"\"!\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Problem 2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 42 "a(n) = 7*a(n-1)-11*a(n-2); a(0)=3,a(1)=-2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#%\"nG,&-F%6#,&F'\"\"\"F,!\"\" \"\"(*&\"#6F,-F%6#,&F'F,\"\"#F-F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $/-%\"aG6#\"\"!\"\"$/-F%6#\"\"\"!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Problem 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "a(n) = a (n-1) + n^3; a(1)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#%\"nG ,&-F%6#,&F'\"\"\"F,!\"\"F,*$)F'\"\"$F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#\"\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "What d oes this problem have to do with" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Sum(k^3,k=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$SumG6$\"%h#*/\"#@;\"\"\"%\"nG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "What does it have to do with" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Sum(k,k=1..n)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* $)-%$SumG6$\"#@/F(;\"\"\"%\"nG\"\"#F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Problem 4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "a( n) = a(n-1) + n^5; a(1)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6 #%\"nG,&-F%6#,&F'\"\"\"F,!\"\"F,*$)F'\"\"&F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#\"\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "What does this problem have to do with" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "Sum(k^5,k=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%$SumG6$\"(,T3%/\"#@;\"\"\"%\"nG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Problem 5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "a(n) = 2 *a(n-1)+11*a(n-2)-12*a(n-3)+5*2^n; a(0)=-2,a(1)=3,a(2)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#%\"nG,*-F%6#,&F'\"\"\"F,!\"\"\"\"#*&\" #6F,-F%6#,&F'F,F.F-F,F,*&\"#7F,-F%6#,&F'F,\"\"$F-F,F-*&\"\"&F,)F.F'F,F ," }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/-%\"aG6#\"\"!!\"#/-F%6#\"\"\"\" \"$/-F%6#\"\"#F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Problem 6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "b(n)=b(n-1)/b(n-2); b(0)=2,b (1)=4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"bG6#%\"nG*&-F%6#,&F'\" \"\"F,!\"\"F,-F%6#,&F'F,\"\"#F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/ -%\"bG6#\"\"!\"\"#/-F%6#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Hint: Try a(n) = log(b(n))." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Compute b(10) from your solutio n and compare with b[10] computed by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "b[0]:=2: b[1]:=4: for k from 2 to 10 do b[k]:=b[k-1]/ b[k-2]; od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Problem 7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Maple will solv e some recursive equations which are of infinite order. For example, t ry" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a(2*n) = a(n)+7; a(1) =0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#,$%\"nG\"\"#,&-F%6#F( \"\"\"\"\"(F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#\"\"\"\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "113 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }