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Writing and Using Explicit Rules for Geometric Sequences

Writing and Using Explicit Rules for Geometric Sequences 1.11 - Solution

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We want to find the equation for the nthn^\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 -5{\color{#FF0000}{\text{-} 5}} and the first term 0.6{\color{#0000FF}{0.6}} to write the explicit rule. an=a1rn1an=0.6(-5)n1\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 a6a_6 we will substitute 66 for nn in our rule.
an=0.6(-5)n1a_n=0.6 \cdot \left(\text{-}5\right)^{n-1}
a6=0.6(-5)61a_{\textcolor{darkorange}{6}}=0.6 \cdot \left( \text{-} 5 \right)^{\textcolor{darkorange}{6}-1}
Evaluate right-hand side
a6=0.6(-5)5a_{6}=0.6 \cdot \left( \text{-} 5 \right)^{5}
a6=0.6(-3125)a_{6}=0.6 \cdot \left( \text{-} 3125 \right)
a6=-1875a_{6} = \text{-}1875