Here's the question in in my book:
Define $(b_n)$ by $b_1=1$, $b_n = a_{n+1}+a_{n-1}$ for $n ≥ 2$. $(b_n)$ is known as the sequence of Lucas numbers.
Prove
(i) $b_n = b_{n-1} + b_{n-2}$ for $n ≥ 3$.
(ii) $a_{2n} = a_nb_n$.
where $(a_n)$ is the Fibonacci sequence of numbers.
I was able to prove the first part but was unable to prove the second using mathematical induction. I tried very hard but couldn't solve it. Kindly suggest a way to prove this.
If you have any proof other than induction, then please do share it as it would be very helpful to know other methods too.
