Geometric Series Sum
Sum of the first n terms of a geometric sequence (r ≠ 1).
Formula
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Example
Sum of 2 + 6 + 18 + 54 (a₁=2, r=3, n=4): S = 2·(1−81)/(1−3) = 80.
How it works
A geometric series is a list of numbers where each term is found by multiplying the previous one by the same constant ratio. This sum formula works by pairing the sequence with a shifted copy of itself, which causes most terms to cancel and leaves only the first term, the ratio, and the number of terms. It only applies when the terms really do follow a fixed multiplicative pattern and the common ratio is not 1.
When to use it
Use this formula when you need the total of a finite geometric pattern, such as repeated growth, depreciation, compound-interest-style sequences, or repeated scaling in algebra problems. It is especially useful when the terms are too many to add one by one.
Worked example, step by step
Find the sum of 5 + 15 + 45 + 135 + 405. Here, a₁ = 5, r = 3, and n = 5. S₅ = a₁(1 - r^n) / (1 - r) S₅ = 5(1 - 3^5) / (1 - 3) S₅ = 5(1 - 243) / (-2) S₅ = 5(-242) / (-2) S₅ = 605 So the sum of the first 5 terms is 605.
Common mistakes to avoid
- Using the formula when the terms do not have a constant ratio
- Forgetting to include the exponent n on r, so using r instead of r^n
- Mixing up the first term a₁ with the common ratio r
Did You Know?
The chessboard-and-rice legend — doubling grains on each square — needs more rice than exists on Earth, thanks to geometric growth.
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