Geometric sequences and series
A geometric sequence multiplies by the same fixed factor — the common ratio — to move from one term to the next: 3, 6, 12, 24 and so on. This calculator returns any term, the sum of the first n terms, and, when the ratio is small enough, the sum of infinitely many terms.
The formulas
The nth term is a1 times r to the power (n minus 1). The sum of the first n terms is a1 times (1 minus r to the n) divided by (1 minus r). When the absolute value of r is below 1 the terms shrink toward zero fast enough that the infinite sum settles on a1 divided by (1 minus r).
When the infinite sum exists
Only if the ratio lies strictly between -1 and 1. With a ratio of 2 the terms keep doubling and the total runs away to infinity; with a ratio of one half they halve each time and converge — the idea behind 1/2 + 1/4 + 1/8 … adding up to 1.
Related tools
When each step adds a fixed amount instead of multiplying, use the arithmetic sequence calculator; for the add-the-last-two pattern, the Fibonacci calculator.
Worked example
Start at 3 with a ratio of 2. The 8th term is 3 times 2 to the 7th = 384, and the first eight terms sum to 765. Because the ratio exceeds 1, there is no finite infinite sum.
FAQ
What if the ratio is 1?
Every term equals the first, so the nth term is a1 and the sum is simply n times a1.
Can the ratio be negative?
Yes. A negative ratio makes the terms alternate in sign, and the infinite sum still converges as long as its absolute value is under 1.