The geometric sum formula calculates the total of a geometric sequence. The formula provides a shortcut for quickly summing all the terms in the sequence without adding each one individually.
A geometric sequence is a list of numbers in which each term is found by multiplying the previous one by a fixed ratio. For example, the sequence 3, 6, 12, 24, 48… has a common ratio of 2. Each number is twice the one before it. The sequence can be finite (with a set number of terms) or infinite (continuing forever).
There are two main forms of the geometric sum formula depending on whether the series has a finite length or is infinite.
Finite Geometric Sum Formula
where:
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Sn is the sum of the first n terms
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a is the first term
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r is the common ratio
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n is the number of terms
Infinite Geometric Sum Formula
The geometric sum formula for an infinite series only works if the absolute value of r is less than 1. If r is 1 or greater (or less than -1), the sequence does not converge, and no finite sum exists.
Example Calculation for Finding the Sum of an Infinite Series
Suppose you want to find the sum of the infinite geometric series:
8 + 4 + 2 + 1 + …
This is a geometric sequence with:
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First term a = 8
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Common ratio r = 1/2
Because the absolute value of r is less than 1, the series converges and we can use the infinite geometric sum formula:
So, the infinite sum of the series is 16. Even though the numbers keep going, they get smaller and smaller, and the total approaches 16.
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