Geometric Sequence Calculator
Find any term and the total of a sequence that grows by a fixed multiplier.
The 8th term (a8)
4,374
a8 = 2 × 3^(8 − 1)
Sum of the first 8 terms
6,560
Using S = a₁ × (1 − rⁿ) / (1 − r)
The sequence starts
2, 6, 18, 54, 162, 486, 1,458, 4,374
How it works
In a geometric sequence you multiply by the same number — the common ratio — at every step rather than adding. Start at 2 with a ratio of 3 and you get 2, 6, 18, 54, growing fast.
The nth term follows aₙ = a₁ × r^(n − 1), so you can leap to the 20th term without writing out the first nineteen. Each step tacks on one more factor of r.
Adding them up uses S = a₁ × (1 − rⁿ) / (1 − r). When the ratio happens to be exactly 1 every term is identical, so the sum is simply a₁ × n, and the calculator handles that case for you.
Frequently asked questions
What's the difference from an arithmetic sequence?
An arithmetic sequence adds a fixed amount each step, while a geometric one multiplies by a fixed factor. That's why geometric sequences balloon (or shrink) so much faster — the growth compounds.
Can the ratio be a fraction or negative?
Definitely. A ratio between -1 and 1 makes the terms shrink toward zero, and a negative ratio flips the sign back and forth, giving an alternating sequence like 3, -6, 12, -24.
Why did I get a dash?
A large ratio raised to a high power can overflow the range a computer number can hold. When the nth term or the sum grows past that limit you'll see a dash — try a smaller ratio or fewer terms.