Find the value of the following expression and round to the nearest integer:
sum, from, n, equals, 1, to, 87, of, 20, left parenthesis, 1, point, 0, 6, right parenthesis, start superscript, n, minus, 1, end superscript
n=1
∑
87
​
 20(1.06) 
n−1

The given expression is the sum of a geometric series. The general form for the sum of a geometric series is:

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

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

In this case:
- \(a = 20\) (the first term when \(n = 1\)),
- \(r = 1.06\) (the common ratio),
- \(N = 87\) (the number of terms).

Now, let's compute the sum of the series using the formula:

\[
S_{87} = 20 \frac{1 - (1.06)^{87}}{1 - 1.06}
\]

I will calculate this step by step.The value of the given expression, rounded to the nearest integer, is **52,693**.

Would you like further details or explanations?

Here are five related questions for further practice:
1. How does the formula for the sum of a geometric series change if \(r > 1\)?
2. What would the sum be if the number of terms, \(N\), were increased to 100?
3. How would the result differ if the common ratio were \(1.05\) instead of \(1.06\)?
4. What happens to the sum as \(N\) approaches infinity if \(r = 0.95\)?
5. How would you find the sum if the first term \(a\) were halved?

**Tip:** The sum of an infinite geometric series converges only when the common ratio \(r\) is less than 1.

Find the value of the following expression and round to the nearest integer: sum, from n equals 1 to 87, of 20(1.06)^(n-1)

Learn how to calculate the sum of a geometric series from n = 1 to 87 with the common ratio 1.06 and the first term 20. The problem involves applying the geometric series formula to find the sum, rounded to the nearest integer. Suitable for Grades 9-12.

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