Chapter 8: Infinite Sequences and Series
Section 8.1: Sequences
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.
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
= 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.
Write the recursion equation with the appropriate Maple syntax.
Press the Enter key.
q≔6 an+2−5 an+1+an=0
Apply the rsolve command and press the Enter key.
Table 8.1.8(a) contains the
task template that, given the general term of a sequence, calculates and graphs its first few members.
First index value
Last index value
plotseq,expr,=.., style=point, symbol=solidcircle, color=red
Table 8.1.8(a) The Sequences task template
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