Geometric Sequence Calculator
A geometric sequence multiplies by the same ratio each step. Enter the first term, the common ratio and a term number to get that term, the sum of the terms up to it and — when the ratio is between −1 and 1 — the sum to infinity.
- Sum of the first n terms (Sₙ)
- 765
- Sum to infinity
- —
- Sequence
- 3, 6, 12, 24, 48, 96, 192, 384
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Formula
How to use it
- Enter the first term.
- Enter the common ratio — divide any term by the one before it.
- Enter which term you want, n.
- Read the nth term, the partial sum and, if it exists, the infinite sum.
Worked examples
3, 6, 12, 24 … the 8th term
- nth term (aₙ)
- 384
- Sum of the first n terms (Sₙ)
- 765
- Sequence
- 3, 6, 12, 24, 48, 96, 192, 384
100, 50, 25 … six terms, and the sum to infinity
- nth term (aₙ)
- 3.125
- Sum of the first n terms (Sₙ)
- 196.875
- Sum to infinity
- 200
When an infinite sum exists
If the ratio is between −1 and 1 the terms shrink toward zero and the total settles on a finite value. 100 + 50 + 25 + … never exceeds 200. The same formula shows that 0.999… = 0.9 ÷ (1 − 0.1) = 1.
With a ratio of 1 or more (or −1 or less) the terms do not shrink and the sum has no limit.
How quickly doubling adds up
One grain of rice on the first square of a chessboard, doubled on each of the 64 squares, totals 2⁶⁴ − 1 = 18,446,744,073,709,551,615 grains. A negative ratio makes the terms alternate in sign: 3, −6, 12, −24.
Questions people ask
What is the 8th term of 3, 6, 12, 24…?
384. With a first term of 3 and a ratio of 2, a₈ = 3 × 2⁷.
What is the sum of the first 8 terms of 3, 6, 12, 24…?
765, from 3 × (2⁸ − 1) ÷ (2 − 1).
What is the sum to infinity of 100, 50, 25…?
200, from 100 ÷ (1 − 0.5).