Prove that $gcd(G_b,b)=1$

Question

I'm doing some exercise on Euclidean space and Euclid's algorithm, but I can't prove the following statement:

NOTE: I'm interested only on the first point (a) and if possible I don't want a direct solution I prefer some hints so that I can work on it on my own.

Answer 1

Hint : Show by strong induction that $G_n=a^{n-1}+bX_n$.

Answer 2

Hint: Prove (a) by induction. If $G_n=aG_{n-1}+bG_{n-2}$ and $\gcd(G_{n-1},b)=1$, what can you say about a common divisor for $G_n$ and $b$?