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Geometric Sequence Calculator

Calculate the nth term and series sum from the first term and common ratio.

How it is calculated

Each term is the previous one multiplied by the common ratio r. With first term a1 and term count n, the nth term is an = a1 × rn−1 and the sum of the first n terms is Sn = a1 × (rn − 1) / (r − 1). When r = 1 every term is equal, so the sum is simply a1 × n. The count n is rounded down to a whole number and must be at least 1. The ratio may be negative (terms alternate in sign) or a fraction (terms shrink). If the result is too large to represent, no answer is returned.

Worked example

Take a1 = 3, r = 2, n = 8. The 8th term is 3 × 27 = 3 × 128 = 384.00. The sum is 3 × (28 − 1) / (2 − 1) = 3 × 255 = 765.00. The sequence runs 3, 6, 12, 24, 48, 96, 192, 384, and adding those eight numbers gives the same 765.00.

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Common mistakes

  • Using n as the exponent: the nth term uses rn−1, not rn, because the first term is not multiplied by anything.
  • Entering a percentage growth such as 5 instead of the ratio 1.05; the ratio must be the multiplier itself.
  • Applying the formula to a sequence that adds a constant (arithmetic) instead of multiplying.
  • Entering a decimal n and expecting a fractional term count; the count is rounded down.

About this calculator

This calculator finds the nth term and the sum of a geometric sequence — a list of numbers where each term is found by multiplying the previous one by a fixed common ratio — using the first term and that common ratio as inputs. Unlike an arithmetic sequence, which grows by a constant amount, a geometric sequence grows (or shrinks) by a constant multiplying factor, which produces either accelerating growth or decay depending on whether the ratio is greater or less than 1.

It's a standard topic in algebra and pre-calculus coursework, where students compute specific terms and partial sums to understand exponential patterns. Beyond the classroom, the same structure models real phenomena with multiplicative, compounding growth or decline: compound interest growing a balance by a fixed percentage each period, population or bacterial growth under ideal conditions, and radioactive decay or depreciation, where a fixed fraction is lost at each step rather than a fixed amount.

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Comments (1)

  • Editorial team

    Pay attention to the size of the ratio. When r lies between -1 and 1 the terms shrink and the sum settles toward a1 / (1 - r) instead of growing without limit: with a1 = 100 and r = 0.5 the sum can never pass 200, however large n gets. A negative r makes the terms alternate in sign. If your sum looks far too big, check whether you typed the ratio as 2 when you meant 0.5.

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