A Sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$

question 19

Essay

A sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined as follows:
$$a _ { 1 } = 3 , \quad \text { and } \quad a _ { k } = 4 a _ { k - 1 } + 2 \quad \text { for all integers } k \geq 2 \text {. }$$
(a) Find $a _ { 1 } , a _ { 2 }$ , and $a _ { 3 }$ .
(b) Supposing that $a _ { 5 } = 4 ^ { 4 } \cdot 3 + 4 ^ { 3 } \cdot 2 + 4 ^ { 2 } \cdot 2 + 4 \cdot 2 + 2$ , find a similar numerical expression for $a _ { 6 }$ by substituting the right-hand side of this equation in place of $a _ { 5 }$ in the equation
$$a _ { 6 } = 4 \cdot a _ { 5 } + 2 .$$
(c) Guess an explicit formula for $a _ { n }$ . Simplify your answer using one of the following reference formulas:
$1 + 2 + 3 + \cdots + n = \frac { n ( n + 1 ) } { 2 }$ for all integers $n \geq 1$ .
$1 + r + r ^ { 2 } + \cdots + r ^ { m } = \frac { r ^ { m + 1 } - 1 } { r - 1 }$ for all integers $m \geq 0$ and all real numbers $r \neq 1$ .

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