How to find the sum of a geometric sequence?

The formula for the sum of a geometric sequence

A geometric sequence multiplies each term by a fixed number called the common ratio, r. When you add these terms together, you get a geometric series. Calculating the total value of these terms comes down to three key numbers: the first term (a_1), the common ratio (r), and the number of terms (n).

The standard formula for the sum of the first n terms is S_n = a_1 * (1 - r^n) / (1 - r), provided that r is not equal to 1. If r is greater than 1, you can also write this as S_n = a_1 * (r^n - 1) / (r - 1) to keep the numerator and denominator positive.

Step-by-step method with numbers

First, find the common ratio r by dividing the second term by the first term. For instance, in the sequence 3, 6, 12, 24, 48, divide 6 by 3 to find r = 2.

Second, identify your first term a_1 and the count n. Here, a_1 = 3 and n = 5.

Third, substitute these numbers into the formula: S_5 = 3 * (1 - 2^5) / (1 - 2). Calculate 2^5 = 32, so 1 - 32 = -31. Then divide by 1 - 2 = -1 to get 31. Finally, multiply by 3 to find S_5 = 93.

Real-world growth and common mistakes

Geometric series model percentage growth over time, such as money in a compound savings account or the spread of an online post. If an account starts with a 100 dollar deposit and grows by 5 percent each period, the common ratio is 1.05.

The most frequent error is calculating (a_1 * r)^n instead of a_1 * (r^n). The exponent applies only to the ratio r. Another common mistake is miscounting n, especially when given the first and last terms rather than the count directly.