Sum of the terms of a geometric sequence

The formula for the sum

To calculate the sum of the first terms of a geometric sequence (u_0, u_1, …, u_n) with a common ratio q not equal to 1, we use:

S = u_0 + u_1 + ... + u_n = u_0 * (1 - q^(n+1)) / (1 - q)

If q = 1, all the terms are equal and the sum is simply S = (n+1) * u_0.

Example

Let’s calculate the sum of the first 5 terms (n = 4) of the sequence u_0 = 1, q = 2:

S = 1 * (1 - 2⁵) / (1 - 2) = (1 - 32) / (-1) = (-31)/(-1) = 31

We can check this by adding the terms directly: 1 + 2 + 4 + 8 + 16 = 31. This confirms the result.

General formula to remember

S = first term * (1 - q^(number of terms)) / (1 - q)

The most common pitfall is getting the number of terms used in the exponent wrong: it corresponds to the NUMBER of terms added together, not to the last index n.

Practical application

A capital sum of 1,000 euros invested each year generates interest which is added to the capital, multiplying it by 1.05 (i.e. +5%) each year. The capital forms a geometric sequence with common ratio q = 1.05, and the sum can be used, for example, to calculate the total accumulated over several years in a savings plan.

Common pitfall

This formula applies ONLY if q is not equal to 1. If q = 1, the general formula would result in division by zero: in this case, one must use S = (number of terms) * u₀.