Geometric Series Sum Calculator: Find the Total of a Sequence

Understanding geometric series is a key part of algebra and calculus. Whether you're dealing with financial growth, population models, or simply solving a homework problem, knowing how to find the sum of a geometric series is incredibly useful. Our Geometric Series Sum Calculator makes this process straightforward, handling both finite and infinite series for you.

Explanation

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, in the series 2, 6, 18, 54., the first term (a) is 2, and the common ratio (r) is 3. The sum of a geometric series is the total value when you add all its terms together. There are two main types: finite geometric series, which have a specific number of terms, and infinite geometric series, which continue indefinitely. For an infinite series to have a finite sum (to 'converge'), the absolute value of its common ratio must be less than 1 (i.e., |r| < 1). If |r| ≥ 1, the series 'diverges', meaning its sum grows infinitely large or oscillates.

Formula

The formulas for calculating the sum of a geometric series are: **For a Finite Geometric Series (n terms):**
S_n = a * (1 - r^n) / (1 - r)
(where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms. This formula applies when r ≠ 1. If r = 1, S_n = a * n) **For an Infinite Geometric Series:**
S_∞ = a / (1 - r)
(This formula applies only when the absolute value of 'r' is less than 1, i.e., |r| < 1.)

Example

Let's consider a practical example. Imagine you're calculating the total distance a bouncing ball travels. If a ball is dropped from 10 meters and bounces back up 0.8 times its previous height, how far does it travel before coming to rest? This is an infinite geometric series problem. Here, the first term (a) for the downward travel is 10m. The subsequent downward travels are 10 * 0.8, 10 * 0.8^2, and so on. The upward travels are 10 * 0.8, 10 * 0.8^2, etc. So, for the total distance, we can consider the initial drop and then two infinite series (one for upward bounces, one for downward bounces after the first). Let's simplify: total distance = initial drop + 2 * (sum of infinite series of bounces). Initial drop = 10m. For the infinite series of bounces: first term (a) = 10 * 0.8 = 8m (first bounce up/down), common ratio (r) = 0.8. Sum of infinite series = a / (1 - r) = 8 / (1 - 0.8) = 8 / 0.2 = 40m. Total distance = 10 + 2 * 40 = 10 + 80 = 90m. Using the calculator, you'd input a=8, r=0.8, and leave 'Number of Terms' blank, then double the result and add the initial drop.

How to use the related calculator

Using our Geometric Series Sum Calculator is straightforward. First, enter the 'First Term (a)' of your series. Next, input the 'Common Ratio (r)'. If you are calculating the sum of a finite series, enter the 'Number of Terms (n)' in the optional field. If you're working with an infinite geometric series that converges, simply leave the 'Number of Terms' field blank. The calculator will instantly display the sum of your series.

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About the author

Muhammad Ali. Muhammad Ali is a full-stack developer and founder of KruskalCode. He builds SaaS platforms and automation systems with React and Laravel, and helps teams ship fast, scalable tools.

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