Question 66
The sequences $\left(a_{n}\right)_{n=1}^{\infty}$ and $\left(b_{n}\right)_{n=1}^{\infty}$ are such that $a_{n+1}=2 b_{n}-a_{n}$ and $b_{n+1}=2 a_{n}-b_{n}$ for every $n$. Prove that:
a) $a_{n+1}=2\left(a_{1}+b_{1}\right)-3 a_{n}$;
b) if $a_{n}\gt 0$ for every $n$, then $a_{1}=b_{1}$.
a) $a_{n+1}=2\left(a_{1}+b_{1}\right)-3 a_{n}$;
b) if $a_{n}\gt 0$ for every $n$, then $a_{1}=b_{1}$.