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{SECT 0 {PARA 256 "" 0 "" {TEXT 256 47 "Explicit and Recursive Definit
ions of Sequences" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0
"" {TEXT -1 9 "Jim Herod" }}{PARA 259 "" 0 "" {TEXT -1 18 "Professor E
meritus" }}{PARA 260 "" 0 "" {TEXT -1 12 "Georgia Tech" }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 93 "Sequences can be defined in many different ways. Two of t
hese might be identified as follows:" }}{PARA 0 "" 0 "" {TEXT -1 82 " \+
(1) an explicit formula, or (2) a recursio
n formula. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 226 "In the first, the nth term of the sequence can be obtained fro
m a formula involving n. In the second, the nth term of the sequence c
an be obtained from the previous term. With this latter method, an int
ial term must be given. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 81 "We illustrate both these methods of defining a seq
uence. We then list five terms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Explicit Definition" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "With this method, \+
we give an explicit formula for the nth term of the sequence " }
{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 42 " by specifying a
formula for the nth term." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0
"" 0 "" {TEXT -1 9 "Define " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG
" }{TEXT -1 18 " as follows: " }{XPPEDIT 18 0 "a[n] = 38+15*n-2*n
^2;" "6#/&%\"aG6#%\"nG,(\"#Q\"\"\"*&\"#:F*F'F*F**&\"\"#F**$F'F.F*!\"\"
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 35 "We list five terms of the sequence." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "a:=n->38+15*n-2*n^2;" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 37 "for i from 1 to 5 do\nprint(a(i));\nod;" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "For many
reasons, it is often of interest to know when a sequence goes above a
certain level, or falls below some level. Here, we ask: When does thi
s sequence become negative?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(a(n)<0,n);" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT
1 {PARA 3 "" 0 "" {TEXT -1 21 "Recursive Definition:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "With this method, we de
fine the " }{XPPEDIT 18 0 "n^th;" "6#)%\"nG%#thG" }{TEXT -1 130 " term
of the sequence from the previous term. Naturally, with this method, \+
we need to give an initial value. We define a sequence " }{XPPEDIT 18
0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT -1 20 " and start it with " }
{XPPEDIT 18 0 "b[0] = 25;" "6#/&%\"bG6#\"\"!\"#D" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
9 "b(0):=25;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "for n from \+
1 to 5 do\nb(n):=(1.07)*b(n-1)-(1.03)^(n-1)*5;\nod;" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 664 "Determining when a seq
uence that is defined recursively goes negative, say, is harder. One m
ight suggest two ways to do this: compute enough terms to discover whe
n the sequence becomes negative or find an explicit formula for the se
quence. The first method is sort of a trial-and-error method. At best,
the sequence becomes negative within the computing time the machine c
an allocate to the problem. At worse, it never goes negative and the c
omputer continues looking for a negative value until smoke rises! On t
he other hand, changing from a recursive definition to an explicit for
mula for the sequence might be hard. We illustrate how Maple can be as
ked to help." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "restart; Dig
its:=5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "recur:=b(n)=(1.0
7)*b(n-1)-(1.03)^(n-1)*5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
26 "rsolve(\{recur,b(0)=25\},b);" }}}{PARA 0 "" 0 "" {TEXT -1 54 "We n
ow have an explicit formula. We define a sequence " }{XPPEDIT 18 0 "c[
n];" "6#&%\"cG6#%\"nG" }{TEXT -1 62 " from this formula, print five te
rms, and ask when it is zero." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 16 "c:=unapply(%,n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f
or i from 1 to 5 do\nprint(evalf(c(i)));\nod;" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 17 "fsolve(c(n)=0,n);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 24 "Exercise for the reader:" }}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{PARA 0 "" 0 "" {TEXT -1 148 "We showed how to change a sequence \+
that is defined recursively into a sequence that is defined explicitly
. Try to do this the other way. A sequence " }{XPPEDIT 18 0 "a[n];" "6
#&%\"aG6#%\"nG" }{TEXT -1 182 " was defined explicitely above. Find a \+
recursive definition for the sequence. Then, ask Maple to find an expl
icit formula for the sequence you made. Do you come back to the origin
al " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 34 "? Here is \+
a hint for how to start." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "r
estart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "a:=n->38+15*n-2*
n^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "a(n+1)-a(n);\nsimpl
ify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
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