https://file.aiquickdraw.com/mathbot/file/1726703927737gwnrmcu8.jpg

The image shows a finite geometric series where the first term is \(5\) and the subsequent terms are fractions, specifically:

\[
5 + \frac{5}{4} + \frac{5}{16} + \frac{5}{64} + \cdots + \frac{5}{16384}
\]

This is a geometric series, and the general form of a geometric series is:

\[
S_n = a + ar + ar^2 + \cdots + ar^{n-1}
\]

Where:
- \(a\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the number of terms in the series.

From the series:
- The first term \(a = 5\),
- The common ratio \(r = \frac{1}{4}\) (as the denominator is multiplied by 4 for each subsequent term),
- The last term in the series is \(\frac{5}{16384}\), so we can calculate the number of terms \(n\).

Let's find the total sum of the series using the formula for the sum of a finite geometric series:

\[
S_n = \frac{a(1 - r^n)}{1 - r}
\]

I'll now calculate the sum of this geometric series.The sum of the given finite geometric series is approximately \(6.667\). The series consists of 8 terms.

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How does the common ratio affect the convergence of a geometric series?
2. What is the difference between a finite and an infinite geometric series?
3. How do you determine if a geometric series converges or diverges?
4. Can the sum formula for a geometric series be applied to any type of series? Why or why not?
5. What happens to the sum of a geometric series if the common ratio is greater than 1?

**Tip**: In a geometric series, when the common ratio \( |r| < 1 \), the series converges, making it easier to find the sum.

Find the sum of the given finite geometric series: 5 + 5/4 + 5/16 + 5/64 + ... + 5/16384.

Learn how to calculate the sum of a finite geometric series with the first term 5 and a common ratio of 1/4. This problem explores the application of the geometric series summation formula, suitable for Grades 10-12.

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