Calculator Geometric Series of N Terms

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 a geometric series with n terms.

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 series can be written as:

a + ar + ar² + ar³ + ... + ar^(n-1)

Where:

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

The sum of the first n terms of a geometric series is called the nth partial sum. The formula for the sum depends on whether the common ratio is 1 or not.

Formula and Calculation

The sum of the first n terms of a geometric series is given by:

If r ≠ 1, the sum S = a(1 - rⁿ)/(1 - r)

If r = 1, the sum S = n × a

This calculator uses these formulas to compute the sum of a geometric series with n terms. The calculation is performed when you click the "Calculate" button.

Note: For the formula to be valid, the common ratio r must be a real number and not equal to 1 when n is infinite. For finite n, r can be any real number except 0.

Worked Examples

Example 1: Common Ratio Not Equal to 1

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

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

The sum of the first 5 terms is 93.

Example 2: Common Ratio Equal to 1

Find the sum of the first 4 terms of a geometric series where the first term a = 5 and the common ratio r = 1.

S = 4 × 5 = 20

The sum of the first 4 terms is 20.

Practical Applications

Geometric series have many practical applications in various fields:

Finance: Calculating the future value of an investment with compound interest
Physics: Modeling radioactive decay
Computer Science: Algorithms and data structures
Engineering: Signal processing and control systems

Understanding geometric series is essential for solving problems in these areas.

FAQ

What is the difference between a geometric series and an arithmetic series?: A geometric series has a common ratio between terms, while an arithmetic series has a common difference between terms.
When is the sum of an infinite geometric series finite?: The sum of an infinite geometric series is finite only when the absolute value of the common ratio is less than 1.
Can the common ratio be negative?: Yes, the common ratio can be negative. This results in alternating signs in the series.
What happens if the common ratio is 1?: If the common ratio is 1, the series becomes an arithmetic series with a common difference of 0, and the sum is simply n times the first term.