Geometric Series Calculator

This geometric series calculator works out the nth term and the running sum of a geometric sequence, a pattern where each number is the one before it multiplied by a fixed ratio, such as 1, 2, 4, 8, 16. You enter three values: the first term (a1), the common ratio (r), and the number of terms (n) you want to add up. The calculator instantly returns the nth term of the sequence, the finite sum of all n terms, and the infinite sum, which only exists when the absolute value of the ratio is less than one. If the ratio is 1 or greater (or minus one or less), the series keeps growing forever, so the infinite sum is shown as diverging rather than a number. This kind of series turns up constantly in real situations: compound interest and investment growth, a ball bouncing to a fraction of its previous height each time, repeated percentage discounts, and population or decay models in science. Enter your own numbers to check working by hand, verify a textbook answer, or explore how quickly a sequence grows or shrinks as the ratio changes. The full formula for the finite and infinite sum, a worked example using a1 = 1, r = 2 and n = 5, and answers to common questions on when a geometric series converges are set out below.

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First term a1

Common ratio r

Number of terms n

Sum = 31

nth term16

Infinite sum (if |r|<1)diverges

The formula

The nth term is a_n = a1 r^(n minus 1). The sum of n terms is S = a1 (1 minus r^n) / (1 minus r) for r not equal to 1, or a1 times n if r is 1. If the absolute value of r is less than 1, the infinite sum is a1 / (1 minus r).

Worked example

For first term 1, ratio 2, five terms (1 + 2 + 4 + 8 + 16), the fifth term is 16 and the sum is 31. Since r is 2, the infinite series diverges. Enter 1, 2, 5 to confirm.

Frequently asked questions

What is a geometric series?

The sum of a geometric sequence, where each term is the previous times a constant ratio r.

When does the infinite sum exist?

Only when the absolute value of r is less than 1. Then the infinite sum is a1 divided by (1 minus r).

What is the finite sum formula?

a1 times (1 minus r to the n) over (1 minus r), valid whenever r is not 1.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.