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.