- #1
pupeye11
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Homework Statement
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]
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?