A Sequence Is Defined Recursively as Follows $$a _ { 0 } = 2 \quad \text { and } \quad a _ { k } = 4 a _ { k - 1 } + 1 \quad \text { for all } k \geq 1 .$$

Essay

A sequence is defined recursively as follows: $$a _ { 0 } = 2 \quad \text { and } \quad a _ { k } = 4 a _ { k - 1 } + 1 \quad \text { for all } k \geq 1 .$$
It is proposed that an explicit formula for this sequence is $$a _ { n } = \frac { 7 \cdot 4 ^ { n } - 1 } { 3 }$$
Use mathematical induction to check whether this proposed formula is correct.