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The recursive definition for the geometric sequence with initial term \(a\) and common ratio \(r\) is \(a_n = a_{n-1}\cdot r; a_0 = a\text{.}\) To get the next term we multiply the previous term by \(r\text{.}\) We can find the closed formula like we did for the arithmetic progression. Write

\begin{align*} a_0 \amp = a\\ a_1 \amp = a_0\cdot r\\ a_2 \amp = a_1 \cdot r = a_0\cdot r\cdot r = a_0\cdot r^2\\ \amp \vdots \end{align*}

We must multiply the first term \(a\) by \(r\) a number of times, \(n\) times to be precise. We get \(a_n = a\cdot r^{n}\text{.}\)

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