What is the approximate sum of this series?
$\sum_{k=1}^8 5\left(\frac{4}{3}\right)^{(k-1)}$
The problem asks for the sum of a finite geometric series, which is given by the expression \(\sum_{k=1}^8 5\left(\frac{4}{3}\right)^{(k-1)}\). A finite geometric series has the general form \(\sum_{k=1}^n a r^{k-1}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. Comparing our series to this general form, we can identify:
The formula for the sum of a finite geometric series with \(r
eq 1\) is:
\[S_n = a \frac{r^n - 1}{r - 1}\]
Substituting the identified values \(a = 5\), \(r = \frac{4}{3}\), and \(n = 8\), we get:
\[S_8 = 5 \cdot \frac{\left(\frac{4}{3}\right)^8 - 1}{\frac{4}{3} - 1}\]
First, we simplify the denominator:
\[\frac{4}{3} - 1 = \frac{1}{3}\]
Substituting this back into the formula gives:
\[S_8 = 5 \cdot \frac{\left(\frac{4}{3}\right)^8 - 1}{\frac{1}{3}}\]
This simplifies to:
\[S_8 = 15 \cdot \left(\left(\frac{4}{3}\right)^8 - 1\right)\]
Using the value \(\left(\frac{4}{3}\right)^8 \approx 9.9887\), the calculation becomes:
\[S_8 \approx 15 \cdot (9.9887 - 1) = 15 \cdot 8.9887 \approx 134.83\]
Thus, the sum is approximately \(134.83\).
Geometric series are widely used in finance to calculate the future value of a series of payments. For instance, if you save an amount that increases by a fixed percentage (common ratio) each month, you can model the total accumulated savings over a specific number of months using a geometric series sum formula. This allows you to project your financial growth over time and make informed budgeting decisions.