Find The Sum of The Following Geometric Series Calculator

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This calculator helps you find the sum of such a series quickly and accurately.

What is a Geometric Series?

A geometric series is a series where each term after the first is found by multiplying the previous term by a constant called the common ratio (r). The general form of a geometric series is:

a + ar + ar² + ar³ + ... + arⁿ

Where:

a is the first term
r is the common ratio
n is the number of terms

Geometric series are common in finance, physics, and engineering. They can be finite (with a specific number of terms) or infinite (extending to infinity).

The Formula for Sum of Geometric Series

The sum of a finite geometric series can be calculated using the following formula:

Sₙ = a(1 - rⁿ) / (1 - r) when r ≠ 1

Sₙ = n*a when r = 1

Where:

Sₙ is the sum of the first n terms
a is the first term
r is the common ratio
n is the number of terms

For an infinite geometric series (where n approaches infinity), the sum converges only if |r| < 1:

S = a / (1 - r) when |r| < 1

Note: The calculator handles both finite and infinite series cases automatically based on your input.

How to Use the Calculator

Enter the first term (a) of your geometric series
Enter the common ratio (r) between terms
Specify whether you want to calculate a finite series (enter number of terms) or an infinite series (check the box)
Click "Calculate" to get the sum
Review the result and explanation

The calculator will display the sum with a clear explanation of how it was calculated and what the result means.

Worked Examples

Example 1: Finite Geometric Series

Find the sum of the first 5 terms of a geometric series where the first term is 2 and the common ratio is 3.

S₅ = 2(1 - 3⁵) / (1 - 3) = 2(1 - 243) / (-2) = 2(-242)/(-2) = 242

The sum of the first 5 terms is 242.

Example 2: Infinite Geometric Series

Find the sum of an infinite geometric series where the first term is 5 and the common ratio is 0.2.

S = 5 / (1 - 0.2) = 5 / 0.8 = 6.25

The sum of the infinite series is 6.25.

Frequently Asked Questions

What is the difference between arithmetic and geometric series?

An arithmetic series has a constant difference between terms, while a geometric series has a constant ratio between terms. For example, 2, 4, 8, 16 is a geometric series with ratio 2, while 2, 4, 6, 8 is an arithmetic series with difference 2.

When does an infinite geometric series converge?

An infinite geometric series converges (has a finite sum) only when the absolute value of the common ratio is less than 1 (|r| < 1). If |r| ≥ 1, the series diverges and has no finite sum.

Can I use this calculator for financial applications?

Yes, geometric series are commonly used in finance for calculations like future value of annuities, present value of perpetuities, and other time-value-of-money problems.