To find the sum of the first seven terms of a geometric series, we use the formula:

$S_n = t_1 \frac{1 - r^n}{1 - r}$

Where:

• $S_n$ is the sum of the first $n$ terms,
• $t_1$ is the first term of the series,
• $r$ is the common ratio,
• $n$ is the number of terms.

Given:

• $t_1 = 13$,
• $r = 5$,
• $n = 7$.

Let's substitute the values into the formula:

$S_7 = 13 \times \frac{1 - 5^7}{1 - 5}$

First, calculate $5^7$:

$5^7 = 78125$

Now substitute:

$S_7 = 13 \times \frac{1 - 78125}{1 - 5}$

$S_7 = 13 \times \frac{-78124}{-4}$

Simplify:

$S_7 = 13 \times 19531$

Now multiply:

$S_7 = 253903$

Thus, the sum of the first seven terms of the geometric series is $\boxed{253903}$.

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. How do you find the sum of an infinite geometric series?
2. What happens to the sum of the geometric series if the common ratio $r$ is less than 1?
3. Can you derive the sum formula for a geometric series?
4. How does the number of terms affect the sum of a geometric series?
5. How can geometric series be applied in real-life scenarios?

Tip: When the common ratio $r$ is between -1 and