Find the nth term of a geometric sequence — one where each term is a fixed multiple of the one before it.
How it works
The formula is aₙ = a₁ × r^(n−1). Starting at 3 with a common ratio of 2, the 5th term is 3 × 2⁴ = 48.
What this does not include
This finds a single term’s value. For the sum of a range of terms in a geometric (or arithmetic) series, see this site’s sum of series calculator instead.
How to use this calculator
- Enter the first term.
- Enter the common ratio.
- Enter which term number you want.
A worked example
First term 3, common ratio 2, 5th term: nth term = a1 × ratio^(n−1) = 3 × 2⁴ = 48.
First term 1, common ratio 0.5, 4th term: 0.125 — a ratio less than 1 produces a shrinking sequence.
What the variables mean
| Variable | Meaning |
|---|---|
| First term (a1) | The sequence’s starting value |
| Common ratio | The fixed factor each term is multiplied by |
| n | Which term number to find |
Edge cases worth knowing
A ratio between 0 and 1 produces a decreasing (but never negative) sequence — unlike an arithmetic sequence with a negative common difference, which can cross zero into negative values.
n must be at least 1 — there’s no “zeroth term” in a standard geometric sequence, so the calculator declines to show a result for n=0.
What happens when the common ratio is between −1 and 1?
The terms shrink toward zero as n increases, since each term is a fraction of the one before — the sequence 1, 0.5, 0.25, 0.125… is a familiar example.
Can the common ratio be negative?
Yes — a negative ratio makes the sequence alternate in sign from term to term, while the magnitude still follows the same geometric pattern.
How is a geometric sequence different from an arithmetic one?
An arithmetic sequence adds a constant amount each step; a geometric sequence multiplies by a constant ratio each step, producing very different growth (or decay) behavior over many terms.
