Activity 8.3.2.
Let \(a = 1\) and \(r = \frac{2}{5}\) be real numbers and consider the corresponding finite geometric series
\begin{equation}
S_n = 1 + \frac{2}{5} + \left(\frac{2}{5}\right)^2 + \cdots + \left(\frac{2}{5}\right)^{n-2} + \left(\frac{2}{5}\right)^{n-1}\text{.}\tag{8.3.2}
\end{equation}
Our goal in this activity is to find a shortcut formula for \(S_n\) that can be written as a single fraction that does not involve a sum of \(n\) terms.
(a)
Multiply both sides of Equation (8.3.2) by \(r = \frac{2}{5}\text{.}\) Write the new equation in the form
\begin{equation}
\frac{2}{5} \cdot S_n = \fillinmath{XXXXX}\text{.}\tag{8.3.3}
\end{equation}
(b)
Now subtract Equation (8.3.3) from Equation (8.3.2), and explain why it follows that
\begin{equation}
S_n - \frac{2}{5} \cdot S_n = 1 - \left(\frac{2}{5}\right)^{n}\text{.}\tag{8.3.5}
\end{equation}
(c)
Solve Equation (8.3.5) for \(S_n\) to find a simple formula for \(S_n\) that does not involve adding \(n\) terms.
(d)
How would your work above change if instead of the original geometric sum \(S_n\text{,}\) we considered the situation with \(a = 7\text{,}\)
\begin{equation*}
S_n = 7 + 7 \cdot \frac{2}{5} + 7 \cdot \left(\frac{2}{5}\right)^2 + \cdots + 7 \cdot \left(\frac{2}{5}\right)^{n-1}\text{?}
\end{equation*}
