HELP!!! Will you please help me with this calculus question? THANK YOU!!!!?

Question

I am so confused with this problem. I don't know how to work it out so that it'll work at all. The problem is this:

Let f1 = 1, f2 = 2, fn = fn-1 + fn-2 (for all n that is greater than or equal to 3)

fn is called the nth Fibonacci number.

a) Show that x/(1-x-x^2) = Summation from n=1 to infinity
(fn x^n)

(Hint: Let g(x) = Summation from n=1 to infinity
(fn x^n). Prove that g(x) = (1-x-x^2) = x.)

Any help to get me started would be greatly appreciated. Thank you so much!!!!!!!!!!!!!!!

Answer 1

I think in the hint that it should be
Prove that g(x).(1-x-x^2) = x.
It also looks as if it should be f2 = 1, not f2 = 2.

g(x).(1-x-x^2)
= Σ(n=1 to ∞) fn x^n (1 - x - x^2)
= Σ(n=1 to ∞) (fn x^n - fn x^(n+1) - fn x^(n+2))
= Σ(n=1 to ∞) fn x^n - Σ(n=2 to ∞) f(n-1) x^n - Σ(n=3 to ∞) f(n-2) x^n
= f1 x + f2 x^2 - f1 x^2 + Σ(n=3 to ∞) x^n (fn - f(n-1) - f(n-2))
= f1 x + (f2 - f1)x^2
since fn - f(n-1) - f(n-2) = 0.
So if f1 = f2 = 1, we have g(x) . (1 - x - x^2) = x
and hence g(x) = x / (1 - x - x^2).

If we have f1 = 1, f2 = 2 then g(x) = x (1 + x) / (1 - x - x^2).