Hello it's me again and this is the last algebra question for a while!

Start with $\displaystyle a_1 = b_1 = 1$, and define

$\displaystyle a_{n+1} = a_n + 2b_n, \:\ b_{n+1} = a_n + b_n \:\ (n \geq\ 20)$

Work out $\displaystyle \frac{a_n}{b_n} $ for $\displaystyle n = \{1,2,4,5,6,\}$. You should find that $\displaystyle \frac{a_n}{b_n} $ is getting closer to $\displaystyle \surd{2}$ as $\displaystyle n$ increases.

Prove by induction that $\displaystyle a_n^2 - 2b_n^2 = (-1)^n$,

and deduce that $\displaystyle \frac{a_n}{b_n} \rightarrow \surd{2} $ as $\displaystyle n \rightarrow \infty$.

I have no problem up to the point of proving the formula by induction but I have no clue as to how to use the formula to make the deduction. Any help would be appreciated.