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Chapter 8: Problem 33

Find the indicated sum. Use the formula for the sum of the first \(n\) terms of a geometric sequence. $$\sum_{i=1}^{10} 5 \cdot 2^{i}$$

Short Answer

Expert verified
The sum of the first 10 terms of this geometric sequence is 10240.

Step by step solution

01

Identify the first term, common ratio, and number of terms

The first term 'a' of the sequence is \(5 \cdot 2 = 10\), the common ratio 'r' is 2, and the number of terms 'n' is 10.
02

Substitute values into the formula

Substitute the values of 'a', 'r', and 'n', we have into the sum formula \( S = 10 \cdot \frac{1- 2^{10}}{1-2}\)
03

Solve the equation

Solve the equation to get the value of S. The result is -10230, but because we are subtracting this quantity in the fraction, the sum \(S = 10240\).

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