A Sequence Is Defined Recursively as Follows $$s _ { k } = 5 s _ { k - 1 } + 1$$

Essay

A sequence is defined recursively as follows: $$s _ { k } = 5 s _ { k - 1 } + 1$$ for all integers $k \geq 1$
$$s _ { 0 } = 1 \text {. }$$
Use mathematical induction to verify that this sequence satisfies the explicit formula $$s _ { n } = \frac { 5 ^ { n + 1 } - 1 } { 4 } \quad \text { for all integers } n \geq 0 \text {. }$$