For the geometric sequence a1 = 3, r = 1/2, what is S6, the sum of the first six terms?

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Multiple Choice

For the geometric sequence a1 = 3, r = 1/2, what is S6, the sum of the first six terms?

Explanation:
Sum of a geometric sequence is found with S_n = a1 (1 - r^n) / (1 - r) when r ≠ 1. With a1 = 3, r = 1/2, and n = 6, compute r^6 = (1/2)^6 = 1/64. Then S6 = 3 * (1 - 1/64) / (1 - 1/2) = 3 * (63/64) / (1/2) = 3 * (63/64) * 2 = 189/32. This matches summing the six terms directly: 3 + 3/2 + 3/4 + 3/8 + 3/16 + 3/32 = 189/32.

Sum of a geometric sequence is found with S_n = a1 (1 - r^n) / (1 - r) when r ≠ 1. With a1 = 3, r = 1/2, and n = 6, compute r^6 = (1/2)^6 = 1/64. Then

S6 = 3 * (1 - 1/64) / (1 - 1/2) = 3 * (63/64) / (1/2) = 3 * (63/64) * 2 = 189/32.

This matches summing the six terms directly: 3 + 3/2 + 3/4 + 3/8 + 3/16 + 3/32 = 189/32.

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