Geometric Sequence Given A1 R and N 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 (r). This calculator helps you find the nth term of a geometric sequence when you know the first term (a₁), the common ratio (r), and the term number (n).

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 sequence is defined by its first term (a₁) and the common ratio (r).

For example, if a₁ = 2 and r = 3, the sequence would be: 2, 6, 18, 54, 162, ...

Geometric sequences are commonly used in finance, physics, and computer science to model exponential growth or decay.

Formula

The nth term of a geometric sequence

The formula to calculate the nth term (aₙ) of a geometric sequence is:

aₙ = a₁ × r^(n-1)

Where:

aₙ = nth term
a₁ = first term
r = common ratio
n = term number

This formula allows you to find any term in the sequence by knowing the first term, the common ratio, and the term number.

How to use this calculator

Enter the first term (a₁) of the geometric sequence.
Enter the common ratio (r) of the geometric sequence.
Enter the term number (n) for which you want to find the value.
Click the "Calculate" button to compute the nth term.
Review the result and the chart showing the sequence terms.

The calculator will display the nth term of the geometric sequence based on the inputs you provide.

Examples

Example 1

Given a geometric sequence with a₁ = 3 and r = 2, find the 5th term (a₅).

Using the formula:

a₅ = 3 × 2^(5-1) = 3 × 16 = 48

The 5th term is 48.

Example 2

Given a geometric sequence with a₁ = 5 and r = 0.5, find the 4th term (a₄).

Using the formula:

a₄ = 5 × 0.5^(4-1) = 5 × 0.125 = 0.625

The 4th term is 0.625.

FAQ

What is the difference between an arithmetic sequence and a geometric sequence?

An arithmetic sequence has a constant difference between consecutive terms, while a geometric sequence has a constant ratio between consecutive terms.

Can the common ratio (r) be negative?

Yes, the common ratio (r) can be negative. This results in a sequence that alternates in sign.

What happens if the common ratio (r) is zero?

If the common ratio (r) is zero, all terms after the first term will be zero.