Geometric Sequence Calculator

Find a geometric sequence’s nth term and finite sum from its starting term, common ratio and term count.

Printed from Calxy · https://www.calxy.net/math/geometric-sequence-calculator

Term 101,024.0000

First term (a₁)

The number the sequence starts with.

Common ratio (r)

The number you multiply by to get from one term to the next.

Number of terms / term index

terms

Which term to find (n). The sum adds up the first n terms.

Term 10

1,024.0000

Sum of first 10 terms2,046.0000

First terms (up to 30)
Index	Term
1	2
2	4
3	8
4	16
5	32
6	64
7	128
8	256
9	512
10	1024

Term indices start at 1. Very large geometric powers can exceed floating-point range; reduce the term count or ratio if the calculator reports an overflow.

How this was calculated

Start at 2 and multiply by the common ratio (the number you multiply by each time), 2, to get each next term.2, 4, 8, 16, …
To jump straight to term 10, multiply the first term by the ratio 9 times (one fewer than n, because the first term is already there).aₙ = a₁r^(n − 1).Put in your numbers: a₁₀ = 2 × 2^(10 − 1) = 2 × 512Answer: 1024
To add up the first 10 terms without listing them all, use the sum shortcut.Sₙ = a₁(1 − rⁿ)/(1 − r), or n × a₁ when r = 1.Put in your numbers: S₁₀ = 2 × (1 − 2^10) ÷ (1 − 2)Answer: 2046

Geometric sequences in simple words

A geometric sequence is a list of numbers where you multiply by the same number each time. That number is called the common ratio.

For example, start at 2 and keep doubling: 2, 4, 8, 16, 32 and so on. The common ratio is 2.

To find the 10th term, multiply the first term by the ratio 9 times: 2 × 2⁹ = 2 × 512 = 1,024.

If the ratio is between −1 and 1, such as 0.5, the numbers shrink towards 0 instead of growing: 16, 8, 4, 2, 1 …

Formula and method

aₙ = a₁r^(n − 1). For r ≠ 1, Sₙ = a₁(1 − rⁿ)/(1 − r). For r = 1, Sₙ = na₁.

Worked example

Starting at 2 with ratio 2, the tenth term is 1,024 and the first ten terms sum to 2,046.

Frequently asked questions

What happens when the ratio is one?

Every term equals the first term, so the sum is the first term multiplied by the count. The calculator handles this without dividing by zero.

Last updated October 19, 2026. Results are estimates for informational purposes only.