Geometric Sequence Calculator
Find the nth term and sum of a geometric sequence.
Result
How it is calculated
You supply the first term a, the common ratio r and the number of terms n. The tool returns the last of those terms and their total: an = a × rn−1 and Sn = a × (1 − rn) / (1 − r). When r is exactly 1 the division would fail, so the engine switches to Sn = a × n.
The number of terms is rounded to the nearest whole number, and anything below 1 gives no result. The ratio may be negative or fractional, and the first term may be negative. Both outputs are shown with four decimals. The tool does not compute the infinite sum.
Worked example
Start at a = 3, double each time (r = 2), and take n = 8 terms. The 8th term is 3 × 27 = 3 × 128 = 384.0000. The sum is 3 × (1 − 28) / (1 − 2) = 3 × (−255) / (−1) = 765.0000. As a check, the terms 3, 6, 12, 24, 48, 96, 192, 384 do add up to 765.
Common mistakes
- Using n as the exponent. The nth term uses rn−1, because the first term has no multiplication yet.
- Entering a percentage growth rate as the ratio. 5% growth per step is r = 1.05, not 5 or 0.05.
- Typing a decimal number of terms: it is rounded, so 7.6 is treated as 8.
- Expecting a result for zero or negative terms; the tool returns nothing below n = 1.
- Confusing the nth term with the sum; the two outputs answer different questions and differ greatly when |r| > 1.
About this calculator
In a geometric sequence, each term is the previous one multiplied by a constant ratio r. The nth term is aₙ = a₁·r^(n−1), and the sum of the first n terms is Sₙ = a₁·(1 − rⁿ)/(1 − r) for r ≠ 1 (with the special case Sₙ = a₁·n when r = 1). When |r| < 1, the sum of the infinite series converges to S = a₁/(1 − r) as n grows without bound.
Students use this to work through algebra and precalculus problems on sequences and series, while it also models real growth-and-decay processes: compound interest and investment growth, bacterial population doubling, radioactive decay (as a decreasing geometric sequence), and the branching structure behind algorithm complexity analysis in computer science. Anyone modeling a quantity that grows or shrinks by a fixed percentage each step — from a biologist tracking a culture to a programmer estimating recursive call counts — relies on this formula.
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