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# Writing and Using Explicit Rules for Geometric Sequences

## Writing and Using Explicit Rules for Geometric Sequences 1.8 - Solution

We see that there is a common ratio $({\color{#FF0000}{2}})$ between $f(x)\text{-}$values making it a geometric sequence.

The explicit rule of a geometric sequence is given below. \begin{aligned} a_n={\color{#0000FF}{a_1}} \cdot {\color{#FF0000}{r}}^{n-1} \end{aligned} Here, ${\color{#0000FF}{a_1}}$ is the initial value, ${\color{#FF0000}{r}}$ is the common ratio. The initial value is ${\color{#0000FF}{8}},$ and the common ratio is ${\color{#FF0000}{2}}.$ Let's substitute the values. \begin{aligned} a_n={\color{#0000FF}{8}} \cdot {\color{#FF0000}{2}}^{n-1} \end{aligned} We want to write a function representing the sequence. To do so, we will substitute ${\color{#009600}{f(x)}}$ for ${\color{#009600}{a_n}}$ and $\textcolor{darkviolet}{x}$ for $\textcolor{darkviolet}{n}.$ \begin{aligned} {\color{#009600}{a_n}}={\color{#0000FF}{8}} \cdot {\color{#FF0000}{2}}^{\textcolor{darkviolet}{n}-1} \quad \Leftrightarrow \quad {\color{#009600}{f(x)}}={\color{#0000FF}{8}} \cdot {\color{#FF0000}{2}}^{\textcolor{darkviolet}{x}-1} \end{aligned} The difference between $a_n$ and $f(x)$ is that they have different domains. The domain of the function is all real numbers whereas the domain of the sequence is the positive integers greater than or equal to $1.$ To graph the function, let's make a table of values.

$x$ $8 \cdot 2^{x-1}$ $f(x)=8 \cdot 2^{x-1}$
$\textcolor{darkviolet}{1}$ $8 \cdot 2^{\textcolor{darkviolet}{1}-1}$ ${\color{#009600}{8}}$
$\textcolor{darkviolet}{2}$ $8 \cdot 2^{\textcolor{darkviolet}{2}-1}$ ${\color{#009600}{16}}$
$\textcolor{darkviolet}{3}$ $8 \cdot 2^{\textcolor{darkviolet}{3}-1}$ ${\color{#009600}{32}}$
$\textcolor{darkviolet}{4}$ $8 \cdot 2^{\textcolor{darkviolet}{4}-1}$ ${\color{#009600}{64}}$

The points $(\textcolor{darkviolet}{1},{\color{#009600}{8}}),$ $(\textcolor{darkviolet}{2},{\color{#009600}{16}}),$ $(\textcolor{darkviolet}{3},{\color{#009600}{32}}),$ and $(\textcolor{darkviolet}{4},{\color{#009600}{64}})$ all lie on the function $f(x).$ Let's plot and connect them with a smooth curve.