Problem 33
Question
Find the indicated sum. Use the formula for the sum of the first \(n\) terms of a geometric sequence. $$\sum_{i=1}^{10} 5 \cdot 2^{i}$$
Step-by-Step Solution
Verified Answer
The sum of the first 10 terms of this geometric sequence is 10240.
1Step 1: 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.
2Step 2: Substitute values into the formula
Substitute the values of 'a', 'r', and 'n', we have into the sum formula \( S = 10 \cdot \frac{1- 2^{10}}{1-2}\)
3Step 3: 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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