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Activity 8.2.2.
Let \(a\) and \(r\) be real numbers (with \(r \ne 1\)) and let
\begin{equation*}
S_n = a+ar+ar^2 + \cdots + ar^{n-1}\text{.}
\end{equation*}
In this activity we will find a shortcut formula for
\(S_n\) that does not involve a sum of
\(n\) terms.
(a)
Multiply
\(S_n\) by
\(r\text{.}\) What does the resulting sum look like?
(b)
Subtract \(rS_n\) from \(S_n\) and explain why
\begin{equation*}
S_n - rS_n = a - ar^n\text{.}
\end{equation*}
(c)
Solve the equation given in the previous step for
\(S_n\) to find a simple formula for
\(S_n\) that does not involve adding
\(n\) terms.
