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Recurrence Relation

by pupeye11
Tags: recurrence, relation
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pupeye11
#1
Jun15-10, 08:51 PM
P: 100
1. The problem statement, all variables and given/known data

The sequence [tex]f_n[/tex] is defined by [tex]f_0=1, f_1=2[/tex] and [tex]f_n=-2f_{n-1}+15f_{n-2}[/tex] when [tex]n \geq 2[/tex]. Let

[tex]
F(x)= \sum_{n \geq 2}f_nx^n
[/tex]

be the generating function for the sequence [tex]f_0,f_1,...,f_n,...[/tex]

Find polynomials P(x) and Q(x) such that

[tex]
F(x)=\frac{P(x)}{Q(x)}
[/tex]

3. The attempt at a solution

[tex]
f_n+2f_{n-1}-15f_{n-2}=0
[/tex]

So since we know that [tex]F(x)=f_0+f_1x+f_2x^2+...+f_nx^n+...[/tex]

[tex]
F(x)=f_0+f_1x+f_2x^2+...+f_nx^n+...
[/tex]

[tex]
2xF(x)=2f_0x+2f_1x^2+...+2f_{n-1}x^n+...
[/tex]

[tex]
-15x^2F(x)= -15f_0x^2-...-15f_{n-2}x^n-...
[/tex]

Summing these I get

[tex]
(1+2x-15x^2)F(x)=f_0+(f_1+2f_0)x+(f_2+2f_1-15f_0)x^2+...+(f_n+2f_{n-1}-15f_{n-2})x^n
[/tex]

After some algebra and substituting [tex]f_0=1, f_1=2[/tex] I get

[tex]
F(x)=\frac{1+4x}{1+2x-15x^2}
[/tex]

So

[tex]
P(x)=1+4x
[/tex]

and

[tex]
Q(x)=1+2x-15x^2
[/tex]

Is this correct?
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lanedance
#2
Jun21-10, 06:59 AM
HW Helper
P: 3,307
looks reasonable to me, you can always check by diferntiating you function a few times


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