Question 3
The sequence $a_1,a_2,a_3,\ldots$ is defined by $a_1=1$, $a_2=2$, and $a_{n+2}=3a_{n+1}-a_n$. The sequence $b_1,b_2,b_3,\ldots$ is defined by $b_1=1$, $b_2=4$, and $b_{n+2}=3b_{n+1}-b_n$. Prove that positive integers $a,b$ satisfy $5a^2-b^2=4$ if and only if $a=a_n$ and $b=b_n$ for some positive integer $n$.