USAJMO 2011 Problem 4

A word is defined as any finite string of letters. A word is a palindrome if it reads the same backwards and forwards. Let a sequence of words $W_0, W_1, W_2,...$ be defined as follows: $W_0 = a, W_1 = b$ , and for $n \ge 2$ , $W_n$ is the word formed by writing $W_{n-2}$ followed by $W_{n-1}$ . Prove that for any $n \ge 1$ , the word formed by writing $W_1, W_2, W_3,..., W_n$ in succession is a palindrome.