Evaluate The Following Geometric Sum Calculator

A geometric sum is the sum of terms in a geometric sequence, where each term after the first is found by multiplying the previous term by a constant called the common ratio. This calculator helps you evaluate geometric sums quickly and accurately.

What is a Geometric Sum?

A geometric sum refers to the total of all terms in a geometric sequence. A geometric sequence 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 (r).

For example, the sequence 2, 6, 18, 54, ... is a geometric sequence where the first term (a) is 2 and the common ratio (r) is 3.

Key Characteristics

First term (a): The starting value of the sequence
Common ratio (r): The factor by which each term is multiplied to get the next term
Number of terms (n): The count of terms in the sequence

How to Calculate Geometric Sum

Calculating a geometric sum involves using the geometric series formula. Here's a step-by-step guide:

Identify the first term (a) of the sequence
Determine the common ratio (r) between consecutive terms
Count the number of terms (n) in the sequence
Apply the geometric series formula to calculate the sum

For finite geometric sequences (where n is known), you can use the formula. For infinite geometric sequences (where n approaches infinity), the series converges only if the absolute value of the common ratio is less than 1.

The Formula

Finite Geometric Series Formula

For a finite geometric series with n terms:

Sₙ = a(1 - rⁿ) / (1 - r)

Where:

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

Infinite Geometric Series Formula

For an infinite geometric series (when |r| < 1):

S = a / (1 - r)

Where:

S = sum of the infinite series
a = first term
r = common ratio (|r| < 1)

The formulas account for the fact that each term is r times the previous term, creating a pattern that can be summed using these mathematical expressions.

Worked Examples

Example 1: Finite Geometric Series

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

Using the formula:

S₅ = 3(1 - 2⁵) / (1 - 2) = 3(1 - 32) / (-1) = 3(-31) / (-1) = 93 / 1 = 93

The sum of the first 5 terms is 93.

Example 2: Infinite Geometric Series

Calculate the sum of an infinite geometric series where the first term is 5 and the common ratio is 0.5.

Using the formula:

S = 5 / (1 - 0.5) = 5 / 0.5 = 10

The sum of the infinite series is 10.

Comparison of Geometric Sum Calculations
Example	First Term (a)	Common Ratio (r)	Number of Terms (n)	Sum
Finite Series	3	2	5	93
Infinite Series	5	0.5	∞	10

FAQ

What is the difference between arithmetic and geometric sums?

An arithmetic sum is the sum of terms in an arithmetic sequence where each term increases by a constant difference. A geometric sum is the sum of terms in a geometric sequence where each term is multiplied by a constant ratio.

When does an infinite geometric series converge?

An infinite geometric series converges only when the absolute value of the common ratio is less than 1 (|r| < 1). If |r| ≥ 1, the series diverges and does not have a finite sum.

Can I use this calculator for complex numbers?

This calculator is designed for real numbers. For complex numbers, you would need a more advanced mathematical tool that handles complex arithmetic.