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# Writing and Using Explicit Rules for Geometric Sequences

## Writing and Using Explicit Rules for Geometric Sequences 1.11 - Solution

We want to find the equation for the $n^\text{th}$ term and the value of the sixth term of the given geometric sequence. First we will find its explicit rule. Let's start by calculating the common ratio.

We will now use the common ratio ${\color{#FF0000}{\text{-} 5}}$ and the first term ${\color{#0000FF}{0.6}}$ to write the explicit rule. $\begin{gathered} a_n=a_1 \cdot r^{n-1} \quad \Leftrightarrow \quad a_n={\color{#0000FF}{0.6}} \cdot \left( {\color{#FF0000}{\text{-} 5}} \right)^{n-1} \end{gathered}$ Finally, to calculate the value of $a_6$ we will substitute $6$ for $n$ in our rule.
$a_n=0.6 \cdot \left(\text{-}5\right)^{n-1}$
$a_{\textcolor{darkorange}{6}}=0.6 \cdot \left( \text{-} 5 \right)^{\textcolor{darkorange}{6}-1}$
Evaluate right-hand side
$a_{6}=0.6 \cdot \left( \text{-} 5 \right)^{5}$
$a_{6}=0.6 \cdot \left( \text{-} 3125 \right)$
$a_{6} = \text{-}1875$