Find the value of the following expression and round to the nearest integer:
sum, from, n, equals, 1, to, 22, of, 500, left parenthesis, 0, point, 8, 4, right parenthesis, to the power n
n=1
∑
22
​
500(0.84)
n

The given expression is a geometric series of the form:

\[
S = \sum_{n=1}^{22} 500 \times (0.84)^n
\]

In a geometric series, the sum \( S \) of the first \( n \) terms can be calculated using the formula:

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

Where:
- \( a \) is the first term of the series.
- \( r \) is the common ratio.
- \( n \) is the number of terms.

For this series:
- \( a = 500 \times 0.84 \) (since the first term occurs at \( n = 1 \))
- \( r = 0.84 \)
- \( n = 22 \)

Let's calculate the sum \( S_{22} \) and round it to the nearest integer.The value of the sum \( \sum_{n=1}^{22} 500 \times (0.84)^n \) rounded to the nearest integer is **2568**.

Would you like more details or have any questions about this solution?

Here are 5 related questions you might find interesting:
1. How do you derive the formula for the sum of a geometric series?
2. What happens to the sum of a geometric series as the number of terms increases?
3. How does the common ratio affect the sum of a geometric series?
4. What is the sum of an infinite geometric series when the common ratio is less than 1?
5. How can you find the sum of a geometric series if the series starts at \( n = 0 \)?

**Tip:** In geometric series, if the absolute value of the common ratio is less than 1, the series converges as the number of terms increases.

Find the value of the following expression and round to the nearest integer: sum, from, n, equals, 1, to, 22, of, 500, left parenthesis, 0, point, 8, 4, right parenthesis, to the power n

Learn how to calculate the sum of a geometric series with 22 terms and a common ratio of 0.84. Using the geometric series formula, find the value and round to the nearest integer. This problem covers exponential decay and the geometric series sum theorem, suitable for students in Grades 9-12.

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