The first term of a geometric sequence is 4 and the common ratio is -2. What is the sum of the first 5 terms?

Question

Accepted Answer

44. Using the geometric series sum formula S_n = a_1(1 - r^n)/(1 - r) with a_1 = 4, r = -2, and n = 5: S_5 = 4(1 - (-2)^5)/(1 - (-2)) = 4(1 - (-32))/(1 + 2) = 4(33)/3 = 4(11) = 44.

We can verify by listing the terms and adding: a_1 = 4, a_2 = 4(-2) = -8, a_3 = -8(-2) = 16, a_4 = 16(-2) = -32, a_5 = -32(-2) = 64. The sum is 4 + (-8) + 16 + (-32) + 64 = 4 - 8 + 16 - 32 + 64 = 44. This confirms our formula result. Note how the terms alternate in sign due to the negative common ratio.

Geometric series with negative ratios are tested on the CLEP exam and require extra care with signs. When r is negative, the terms alternate between positive and negative. The sum formula S_n = a_1(1 - r^n)/(1 - r) works for any r != 1. Pay attention to (-2)^5 = -32 (odd power of a negative is negative) versus (-2)^4 = 16 (even power is positive). Always verify your answer by computing at least the first few terms and checking that the sum makes sense.

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