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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Definitions:

Brachial Artery

The major blood vessel of the upper arm that supplies blood to the arm and hand.

Axillary Artery

A major blood vessel that carries oxygenated blood to the lateral aspect of the torso, armpit, and upper limb.

Median Cubital Artery

The median cubital artery is a superficial blood vessel in the elbow region that connects the cephalic and basilic veins and is often used for venipuncture.

Ulnar Artery

The artery that runs along the side of the forearm that is closest to the body when in an anatomical position, supplying blood to the forearm and hand.