Find the sum of the geometric series with a1 = 5, r = 1/2, and n = 6.

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Multiple Choice

Find the sum of the geometric series with a1 = 5, r = 1/2, and n = 6.

Explanation:
The sum of a finite geometric series is found using S_n = a1 (1 − r^n) / (1 − r) when r ≠ 1. Here, a1 = 5, r = 1/2, and n = 6. Compute r^n = (1/2)^6 = 1/64, so 1 − r^n = 63/64. Also 1 − r = 1 − 1/2 = 1/2. Then S_n = 5 · (63/64) ÷ (1/2) = 5 · (63/64) · 2 = 5 · 63/32 = 315/32. Final sum: 315/32.

The sum of a finite geometric series is found using S_n = a1 (1 − r^n) / (1 − r) when r ≠ 1. Here, a1 = 5, r = 1/2, and n = 6.

Compute r^n = (1/2)^6 = 1/64, so 1 − r^n = 63/64. Also 1 − r = 1 − 1/2 = 1/2. Then

S_n = 5 · (63/64) ÷ (1/2) = 5 · (63/64) · 2 = 5 · 63/32 = 315/32.

Final sum: 315/32.

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