Determine the sum of the first seven terms of the geometric series in which (t1= 13 and r = 5)

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 

Determine the sum of the first seven terms of the geometric series in which t1= 13 and r = 5.

Learn how to find the sum of the first seven terms of a geometric series with a first term of 13 and a common ratio of 5. This problem applies the geometric series sum formula, covers exponents, and explores sequences. Suitable for students in Grades 9-12.

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