A Sequence $c _ { 0 } , c _ { 1 } , c _ { 2 } , \ldots$

question 17

Essay

A sequence $c _ { 0 } , c _ { 1 } , c _ { 2 } , \ldots$ is defined as follows:
$$c _ { 0 } = 1 \quad \text { and } \quad c _ { k } = 7 c _ { k - 1 } + 2 \text { for each integer } k \geq 1 \text {. }$$
(a) Find $c _ { 1 }$ and $c _ { 2 }$ .
(b) Simplify the expression
$$7 ^ { n } + 2 \cdot 7 ^ { n - 1 } + \cdots + 2 \cdot 7 ^ { 2 } + 2 \cdot 7 + 2 .$$
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 $r \neq 1$ .
(c) Use iteration to guess an explicit formula for the sequence $c _ { 0 } , c _ { 1 } , c _ { 2 } , \ldots$

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