Finite Geometric Series
Sum of the first n terms of a geometric series.
Formula
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Example
2+4+8+16 = 30
How it works
A geometric series is a list of terms where each new term is found by multiplying the previous one by the same constant ratio r. This formula gives the total of the first n terms without adding them one by one: multiplying the sum by r shifts every term along, and subtracting the two sums makes most terms cancel, leaving a simple expression. It works best when r is not 1, because then the cancellation produces a usable denominator.
When to use it
Use this when quantities grow or shrink by a fixed multiplier, such as compound interest, population models, repeated discounts, or exponential-style sequences in algebra. It is also useful whenever you need the total of a finite number of terms in a pattern like 3, 6, 12, 24, ...
Worked example, step by step
Find the sum of the first 5 terms of 3 + 6 + 12 + 24 + 48 Here, a = 3, r = 2, and n = 5 Use S_n = a(1 - r^n) / (1 - r) S_5 = 3(1 - 2^5) / (1 - 2) S_5 = 3(1 - 32) / (-1) S_5 = 3(-31) / (-1) S_5 = 93 So, the sum of the first 5 terms is 93.
Common mistakes to avoid
- Forgetting that r is the multiplier between terms, not the difference
- Using the wrong first term a, especially when the series does not start at the first visible number
- Dropping parentheses around 1 - r^n, which can lead to sign errors
Did You Know?
The geometric series formula underlies every loan and mortgage calculation ever made.
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