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The image shows the following math problem:

\[
5 + 15 + 45 + \dots
\]

with a task to determine \( S_{15} \), the sum of the first 15 terms of this sequence.

This is a geometric progression (GP), where the first term \( a = 5 \) and the common ratio \( r = 3 \), since \( \frac{15}{5} = 3 \) and \( \frac{45}{15} = 3 \).

The sum of the first \( n \) terms of a geometric series is given by the formula:

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

Substituting \( a = 5 \), \( r = 3 \), and \( n = 15 \) into the formula:

\[
S_{15} = 5 \frac{1 - 3^{15}}{1 - 3}
\]

Let me compute that for you.The sum of the first 15 terms of the geometric series is \( S_{15} = 35,872,265 \).

Would you like further details or have any questions?

Here are some related questions to expand on this:

1. How would you calculate the sum if the common ratio were negative?
2. What happens to the sum of an infinite geometric series when the common ratio is less than 1?
3. How can you generalize this method for any geometric series?
4. What is the sum of the first 20 terms of this geometric series?
5. How does the common ratio affect the growth of the series?

**Tip:** Always check whether the common ratio is greater than or less than 1 to predict how fast the series grows.

Learn how to determine the sum of the first 15 terms of the geometric progression starting with 5, 15, 45... This problem demonstrates the use of the geometric series summation formula and explores how the common ratio affects the sum. Suitable for Grades 9-12.

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