Geometric Sequence Calculator
Find the nth term and the sum of a geometric sequence from its first term and common ratio.
How it works
nth term = a × r^(n−1) · sum of n terms = a × (rⁿ − 1) ÷ (r − 1), or a × n when r = 1
A geometric sequence multiplies by the same fixed ratio r to move from term to term, so its values grow (or shrink) exponentially rather than by steady steps. The nth-term formula reaches any position with a single power of r, and the sum formula telescopes the series into a compact fraction. When the ratio equals 1 every term is identical, so the sum is just a × n.
Worked example
With first term 2 and ratio 3, the terms run 2, 6, 18, 54 … The 8th term is 2 × 3⁷ = 2 × 2187 = 4374. The first eight terms sum to 2 × (3⁸ − 1) ÷ (3 − 1) = 2 × 6560 ÷ 2 = 6560.
Sources: Geometric progression — Wikipedia
Frequently asked questions
What’s the difference from an arithmetic sequence?
Arithmetic sequences add a constant each step; geometric sequences multiply by a constant. 2, 6, 18, 54 triples each time (geometric), while 2, 6, 10, 14 adds 4 each time (arithmetic).
What if the common ratio is between −1 and 1?
The terms shrink toward zero, and an infinite geometric series then converges to a ÷ (1 − r). For example, 1 + ½ + ¼ + … sums to 2.
Why can the numbers get so large?
Geometric growth is exponential, so terms explode quickly — 2 × 3⁷ is already 4,374. Very large results are shown in scientific notation to stay readable.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.