Writing and Using Explicit Rules for Geometric Sequences

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 using the four terms given in the graph, $224,$ $112,$ $56,$ and $28.$

We will now use the common ratio ${\color{#FF0000}{\frac{1}{2}}}$ and the first term ${\color{#0000FF}{224}}$ to write the explicit rule. $\begin{gathered} a_n=a_1 \cdot r^{n-1} \quad \Leftrightarrow \quad a_n={\color{#0000FF}{224}}\left( {\color{#FF0000}{\dfrac{1}{2}}} \right)^{n-1} \end{gathered}$ Finally, to calculate the value of $a_6$ we will substitute $6$ for $n$ in our rule.