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Geometric Series Sum Calculator

Add up the terms of a geometric sequence instantly, no matter how fast it grows or shrinks.

How it is calculated

The sum of the first n terms uses the closed formula for a geometric series, so the terms never have to be listed.

S = a × (1 − rⁿ) ÷ (1 − r) when r ≠ 1, and S = a × n when r = 1

a is the first term, r the common ratio (a factor: 0.9 means each term is 90 % of the previous one) and n the number of terms. The count must be a whole number of at least 1. A ratio of exactly 1 is handled separately, since the formula would divide by zero. The result is shown to 4 decimals. If the sum is too large to represent (for example ratio 2 with 5000 terms), no result is shown.

Worked example

The first term is 100 and each following term is 90 % of the one before, for 20 terms. Power: 0.9^20 ≈ 0.121577. Numerator: 1 − 0.121577 = 0.878423. Denominator: 1 − 0.9 = 0.1. Then 100 × 0.878423 ÷ 0.1. The sum of the 20 terms is 878.4233. For comparison, an endless series with the same ratio would approach 1000.

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Common mistakes

  • Entering 90 instead of 0.9 for a ratio of 90 %. The ratio is a plain factor, so 90 % is 0.9 and +10 % is 1.1.
  • Confusing this with the nth-term tool. This one adds all n terms; the last term alone is a different, much smaller number.
  • Entering n as the number of steps instead of the number of terms. Four terms (a, ar, ar², ar³) means n = 4.
  • Using a ratio of exactly 1 and expecting a formula with a division. The tool treats that case as a constant series, first term × n.
  • Expecting a result for extreme inputs. With a ratio far above 1 and thousands of terms, the sum overflows and the tool shows nothing.

About this calculator

The sum of a finite geometric series — a sequence where each term is multiplied by a constant common ratio r — is given by Sn = a × (1 − r^n) divided by (1 − r), where a is the first term and n is the number of terms; unlike an arithmetic series, the terms of a geometric series can grow explosively when the ratio is greater than 1, or shrink toward zero when the ratio is between 0 and 1.

Geometric series model compound growth and decay throughout finance and science — compound interest and investment growth, population or bacterial growth models, radioactive decay chains, the depreciation of an asset's value by a fixed percentage each year, and the branching pattern of a chain letter or referral program all follow the same underlying formula.

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