USAJMO 2011 Question 4

A word is a finite string of letters; it is a palindrome if it reads identically forwards and backwards. Define $W_0=a$, $W_1=b$ and $W_n=W_{n-2}W_{n-1}$ for $n\ge2$, where juxtaposition means concatenation. Prove that $W_1W_2\cdots W_n$ is a palindrome for every $n\ge1$.

Proposed by Gabriel Carroll (credit in the supplied notes).