Evaluate the sum of the first 8 terms of a geometric series with initial term a0 = 7 and common ratio r = -1/3.

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Multiple Choice

Evaluate the sum of the first 8 terms of a geometric series with initial term a0 = 7 and common ratio r = -1/3.

Explanation:
When summing a finite geometric series, use the formula S_n = a0 (1 - r^n) / (1 - r) for r ≠ 1. Here, a0 = 7, r = -1/3, and n = 8. Compute r^8 = (-1/3)^8 = 1/6561, so 1 - r^8 = 6560/6561. Also 1 - r = 1 - (-1/3) = 4/3. Thus S_8 = 7 * (6560/6561) / (4/3) = 7 * (6560/6561) * (3/4). Multiply: 7 * 6560 * 3 = 137,760, and 6561 * 4 = 26,244, giving S_8 = 137,760 / 26,244. This simplifies by 12 to 11,480 / 2,187, and since 2,187 = 3^7 and 11,480 is not divisible by 3, it cannot be reduced further. Therefore, the sum of the first eight terms is 11,480/2,187, which matches the given value.

When summing a finite geometric series, use the formula S_n = a0 (1 - r^n) / (1 - r) for r ≠ 1. Here, a0 = 7, r = -1/3, and n = 8.

Compute r^8 = (-1/3)^8 = 1/6561, so 1 - r^8 = 6560/6561. Also 1 - r = 1 - (-1/3) = 4/3. Thus

S_8 = 7 * (6560/6561) / (4/3) = 7 * (6560/6561) * (3/4).

Multiply: 7 * 6560 * 3 = 137,760, and 6561 * 4 = 26,244, giving S_8 = 137,760 / 26,244. This simplifies by 12 to 11,480 / 2,187, and since 2,187 = 3^7 and 11,480 is not divisible by 3, it cannot be reduced further.

Therefore, the sum of the first eight terms is 11,480/2,187, which matches the given value.

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