Chapter 8: Infinite Sequences and Series
Section 8.1: Sequences
Example 8.1.8
If a1=1,a2=−1, and 6 an+2−5 an+1+an=0 defines an for n>2, use Maple to find the general term an.
Solution
Mathematical Solution
The sequence is defined by a linear difference (or recursion) equation with constant coefficients. Such equations have solutions in the form of rn for some value(s) of r. Substituting such a "guess" into the equation results in
0
= 6 rn+2− 5 rn+1+rn
= rn6 r2−5 r+1
= rn2 r−13 r−1
from which it follows that r=1/2 or r=1/3 and the general solution of the recursion equation is an=A/2n+B/3n. Applying the two initial conditions a1=1 and a2=−1 gives the two equations A/2+B/3=1 and A/4+B/9=−1, whose solution is A=−16,B=27.
An explicit representation for the general term of the series is then an=27/3n−16/2n, from which it is clear that the limit of the sequence an is zero.
Maple Solution
Write the recursion equation with the appropriate Maple syntax. Press the Enter key.
q≔6 an+2−5 an+1+an=0
6an+2−5an+1+an=0
Apply the rsolve command and press the Enter key.
rsolveq,a1=1,a2=−1,a
2713n−1612n
Table 8.1.8(a) contains the task template that, given the general term of a sequence, calculates and graphs its first few members.
Tools≻Tasks≻Browse: Algebra≻Sequences
Sequences
General term
expr≔27/3n−16/2n
expr:=273n−162n
Index name
n
First index value
1
Last index value
10
Members
seqexpr,=..
1,−1,−1,−23,−718,−23108,−73648,−2273888,−69723328,−2123139968
Graph
plotseq,expr,=.., style=point, symbol=solidcircle, color=red
Table 8.1.8(a) The Sequences task template
<< Previous Example Section 8.1 Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
