Find N Term Geometric Sequence Calculator

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. This calculator helps you find any term in a geometric sequence when you know the first term and the common ratio.

What is 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). The general form of a geometric sequence is:

a, ar, ar², ar³, ..., arⁿ⁻¹

Where:

a = first term
r = common ratio
n = term number

Geometric sequences are commonly seen in nature, finance, and physics. Examples include population growth, compound interest, and wave patterns.

Formula for finding the nth term

The nth term of a geometric sequence can be found using the following formula:

aₙ = a × r^(n-1)

Where:

aₙ = nth term
a = first term
r = common ratio
n = term number

This formula allows you to calculate any term in the sequence when you know the first term and the common ratio.

How to use the calculator

Enter the first term (a) of the geometric sequence.
Enter the common ratio (r) between terms.
Enter the term number (n) you want to find.
Click the "Calculate" button to find the nth term.
The result will appear in the result box below the calculator.

The calculator will show you the exact value of the nth term based on the inputs you provide.

Examples of geometric sequences

Let's look at some examples to understand how geometric sequences work.

Example 1: Simple geometric sequence

Consider a sequence where the first term (a) is 2 and the common ratio (r) is 3.

The sequence would be: 2, 6, 18, 54, 162, ...

Using the formula to find the 5th term (n=5):

a₅ = 2 × 3^(5-1) = 2 × 81 = 162

Example 2: Geometric sequence with decimal ratio

Consider a sequence where the first term (a) is 10 and the common ratio (r) is 0.5.

The sequence would be: 10, 5, 2.5, 1.25, 0.625, ...

Using the formula to find the 4th term (n=4):

a₄ = 10 × 0.5^(4-1) = 10 × 0.125 = 1.25

Example 3: Geometric sequence with negative ratio

Consider a sequence where the first term (a) is 1 and the common ratio (r) is -2.

The sequence would be: 1, -2, 4, -8, 16, ...

Using the formula to find the 6th term (n=6):

a₆ = 1 × (-2)^(6-1) = 1 × 32 = 32

Frequently Asked Questions

What is the difference between arithmetic and geometric sequences?: An arithmetic sequence has a constant difference between terms, while a geometric sequence has a constant ratio between terms.
Can the common ratio be negative?: Yes, the common ratio can be negative, which results in alternating signs in the sequence.
What happens if the common ratio is 1?: If the common ratio is 1, all terms in the sequence will be equal to the first term.
How do I find the common ratio if I know two terms?: You can find the common ratio by dividing the second term by the first term.
Can geometric sequences be used in finance?: Yes, geometric sequences are used in finance for compound interest calculations and investment growth projections.