A Sequence $S _ { 1 } , S _ { 2 } , S _ { 3 }$

Essay

A sequence $S _ { 1 } , S _ { 2 } , S _ { 3 }$ ... is defined recursively as follows: $$\begin{array} { l } s _ { k } = 5 s _ { k - 1 } + \left( s _ { k - 2 } \right) ^ { 2 } \quad \text { for all integers } k \geq 3 \\s _ { 1 } = 4 \\s _ { 2 } = 8\end{array}$$ Use (strong) mathematical induction to prove that sn is divisible by 4 for all integers $$n \geq 1$$