Geometric Sequence Calculator

Calculate the terms and sum of a geometric sequence where each term is found by multiplying the previous term by a constant ratio.

The Two Formulas at Work

A geometric sequence multiplies each term by a fixed ratio r to get the next one: a(n) = a1 x r^(n-1). The sum of the first n terms uses a second formula: S(n) = a1 x (1 - r^n) / (1 - r), except when r = 1, where the sum is simply a1 x n (since every term is identical).

Worked Example: Diverging Sequence

First term 3, ratio 2, 6 terms. The sequence: 3, 6, 12, 24, 48, 96 (each term double the last). Sum: S(6) = 3 x (1 - 2^6)/(1 - 2) = 3 x (1 - 64)/(-1) = 3 x 63 = 189. You can check this by adding directly: 3+6+12+24+48+96 = 189.

Worked Example: Converging Sequence

First term 8, ratio 0.5, 4 terms. The sequence: 8, 4, 2, 1, summing to 15. Because the ratio's absolute value is less than 1 here, the sequence also has a finite sum to infinity: a1/(1-r) = 8/0.5 = 16 - meaning if you kept halving forever, the running total would approach 16 but never exceed it.

When There's No Sum to Infinity

If the ratio's absolute value is 1 or more (as in the first example, where r = 2), the terms keep growing and the sum has no finite limit - the tool states this explicitly rather than attempting to calculate a number that doesn't exist.

Display Limits

Sequences longer than 20 terms only show the first five and last five values, separated by an ellipsis, to keep the page readable. The full sum calculation still runs on all terms regardless of how many are displayed.