Here is some help by way of an example.
\(B_7\) is the set of all bit-strings of length seven which do not have two consecutive ones.
Ex. \(0101001\in B_7\) if we add \(\color{blue}01\) we get \(0101001{\color{blue}01}\in B_9\)
Taking any string from \(B_7\) and affixing \(\color{blue}01\) its right end gives a term in \(B_9\). THINK ABOUT IT.
Likewise, taking any string from \(B_8\) and affixing \(\color{blue}0\) its right end gives a term in \(B_9\). WHY?
Is it true that doing both operations we get ALL of \(\bf{B_9}~\)?
Have we shown that \(B_{n+1}=B_{n+2}+B_{n-1}~\)?