B. Geometric Series
Completion requirements
B. Geometric Series
A geometric series consists of the terms of a geometric sequence added together. In the previous section, you derived one of the formulas used for geometric series.
\[S_n = \frac{{t_1 \left( {r^n - 1} \right)}}{{r - 1}}\, {\rm {where}}\thinspace r \ne 1 \]
This formula works well when you are given the values for \(t_1 \) , \(r \), and \(n \). What about if you are not given \(n \)? By manipulating the formula, you can generate a different version that helps in these situations.
Factoring out one \(r \) from \(r^n \) gives \(r \cdot r^{n - 1}\). Also, recall that \(t_n = t_1 r^{n - 1} \).
To summarize, the two geometric series formulas are:
\[S_n = \frac{{t_1 \left( {r^n - 1} \right)}}{{r - 1}}\, {\rm {where}}\thinspace r \ne 1 \]
\[S_n = \frac{{rt_n - t_1 }}{{r - 1}}, {\rm {where}}\thinspace r \ne 1 \]
\(t_1 \) is the first term
\(r \) is the common ratio
\(t_n \) is the \(n^{th} \) term in the series
\(n \) is the number of terms
\[S_n = \frac{{t_1 \left( {r^n - 1} \right)}}{{r - 1}}\, {\rm {where}}\thinspace r \ne 1 \]
This formula works well when you are given the values for \(t_1 \) , \(r \), and \(n \). What about if you are not given \(n \)? By manipulating the formula, you can generate a different version that helps in these situations.
\[\begin{align}
S_n &= \frac{{t_1 \left( {r^n - 1} \right)}}{{r - 1}} \\
S_n &= \frac{{t_1 r^n - t_1 }}{{r - 1}} \\
S_n &= \frac{{r\left( {t_1 r^{n - 1} } \right) - t_1 }}{{r - 1}} \\
S_n &= \frac{{rt_n - t_1 }}{{r - 1}} \\
\end{align}\]
S_n &= \frac{{t_1 \left( {r^n - 1} \right)}}{{r - 1}} \\
S_n &= \frac{{t_1 r^n - t_1 }}{{r - 1}} \\
S_n &= \frac{{r\left( {t_1 r^{n - 1} } \right) - t_1 }}{{r - 1}} \\
S_n &= \frac{{rt_n - t_1 }}{{r - 1}} \\
\end{align}\]
Factoring out one \(r \) from \(r^n \) gives \(r \cdot r^{n - 1}\). Also, recall that \(t_n = t_1 r^{n - 1} \).
Key Lesson Marker |
Geometric Series
To summarize, the two geometric series formulas are:
\[S_n = \frac{{t_1 \left( {r^n - 1} \right)}}{{r - 1}}\, {\rm {where}}\thinspace r \ne 1 \]
\[S_n = \frac{{rt_n - t_1 }}{{r - 1}}, {\rm {where}}\thinspace r \ne 1 \]
\(t_1 \) is the first term
\(r \) is the common ratio
\(t_n \) is the \(n^{th} \) term in the series
\(n \) is the number of terms