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

---

Definitions:

Proactive Interference

A phenomenon where old memories hinder the recall of newly formed memories.

Remembering

The process by which past experiences are recalled or retrieved from memory.

Prospective Memory

It represents the part of memory that deals with recalling planned activities or intentions at a predetermined time in the future.

Odor

A smell produced by one or more volatilized chemical compounds that are generally found in low concentrations.