Finite GP Sum Formula

Quick answer: For a GP with n terms and r ≠ 1, Sₙ = a(1 − rⁿ)/(1 − r). Equivalently use a(rⁿ − 1)/(r − 1). When r = 1, the sum is na.

Worked example

Find the sum of the first five terms of 2, 6, 18, … .

1. Here a = 2, r = 3, n = 5.
2. S₅ = 2(3⁵ − 1)/(3 − 1).
3. 3⁵ = 243, so S₅ = 2(242)/2 = 242.

Check the result

Adding 2 + 6 + 18 + 54 + 162 gives 242.

Common mistake to avoid

Do not combine the numerator from one version with the denominator from the other. The finite-sum formula works even when the infinite series does not converge.

Try it yourself

1. Sum 8 + 4 + 2 + 1. 2. Sum six terms with a = 5 and r = 1.

Answers and reasoning

1. 8(1 − (1/2)⁴)/(1 − 1/2) = 15. 2. Six equal terms total 6 × 5 = 30.

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