https://file.aiquickdraw.com/mathbot/file/1727217635746vb8dvv70.jpg

The equation in the image is a summation:

\[
\sum_{n=0}^{15} 2 \left( \frac{4}{3} \right)^n
\]

This represents a finite geometric series with:
- First term \( a = 2 \)
- Common ratio \( r = \frac{4}{3} \)
- The summation index \( n \) ranges from 0 to 15 (so there are 16 terms in total).

The formula for the sum \( S \) of a geometric series is:

\[
S = a \frac{1 - r^{N+1}}{1 - r}
\]

Where:
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( N \) is the number of terms minus one (in this case, \( N = 15 \)).

Let's calculate this geometric series sum.The sum of the given geometric series is approximately \( 592.65 \).

Would you like any further details or have any questions? Here are a few related questions to explore further:

1. How does the common ratio affect the growth of a geometric series?
2. What happens if the common ratio \( r \) is less than 1 in a geometric series?
3. Can you derive the formula for the sum of an infinite geometric series?
4. How would the result change if the first term were different?
5. What real-world scenarios can be modeled by geometric series?

**Tip:** In geometric series, if the common ratio \( |r| < 1 \), the sum can converge to a finite value even if the series is infinite.

Evaluate the summation of the geometric series: Σ from n=0 to 15 of 2*(4/3)^n.

This problem evaluates the sum of a geometric series Σ from n=0 to 15 of 2*(4/3)^n. It covers key concepts such as the sum of a finite geometric series, with the first term a = 2 and common ratio r = 4/3. Learn how to use the geometric series formula to compute the total sum, approximately 592.65.

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