Generating Function (Finding An)

__Discrete Mathematics with Graph Theory__

*by Goodaire and Parmenter*

Page 182 # 10b

**a**_{n} = -10a_{n-1} - 25a_{n-2}, n ≥ 2, given a_{0}=1, a_{1} = 25

So far I have:

f(x) - a_{0} -a_{1}x = -10x(f(x) - a_{0}) - 25x^{2}f(x)

f(x) - 1 -25x = -10x(f(x) - 1) - 25x^{2}f(x)

Then solving for f(x)

I get:

f(x) = (1+35x)/(1+10X+25x^{2})

f(x) = (1+35x)/(1+5x)^{2}

Then when I do partial fractions I get:

(1+35x) = (A/(5x+1)^2) + (B/(5x+1))

Then solving for A & B I get:

A = 7

B = -6

And now I'm having trouble finding my a_{n} coefficient.

Re: Generating Function (Finding An)

Hello, ThatPinkSock!

I am unfamiliar with your method.

I'll show you the procedure I was taught.

We have: .

Let:

Substitute: .

Divide by

The function has the form: .

Substitute the first two terms:

. .

. .

. . . . . . . . . . .

Therefore: .

. . . . . . . . . .

Re: Generating Function (Finding An)

Hi ThatPinkSock,

For the generating function approach, use the series

to expand

and use the series to expand .

Here's one part:

I'll leave the rest for you...