Use Iteration to Find an Explicit Formula for the Sequence $b _ { 0 } , b _ { 1 } , b _ { 2 } , \ldots$

question 33

Essay

Use iteration to find an explicit formula for the sequence $b _ { 0 } , b _ { 1 } , b _ { 2 } , \ldots$ defined recursively as follows:
$$\begin{array} { l } b _ { k } = 2 b _ { k - 1 } + 3 \quad \text { for all integers } k \geq 1 \\b _ { 0 } = 1 .\end{array}$$
If appropriate, simplify your answer using one of the following reference formulas:
(a) $1 + 2 + 3 + \cdots + n = \frac { n ( n + 1 ) } { 2 }$ for all integers $n \geq 1$ .
(b) $1 + r + r ^ { 2 } + \cdots + r ^ { m } = \frac { r ^ { m + 1 } - 1 } { r - 1 }$ for all integers $m \geq 0$ and all real numbers $r \neq 1$ .

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