Explicit Formula for Geometric Sequence for 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. The explicit formula allows you to find any term in the sequence without calculating all the previous terms.

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 is the first term
r is the common ratio
n is the term number

Geometric sequences are common in many areas of mathematics and science, including finance, physics, and computer science.

Explicit Formula for Geometric Sequence

The explicit formula for finding the nth term of a geometric sequence is:

aₙ = a × r^(n-1)

Where:

aₙ is the nth term
a is the first term
r is the common ratio
n is the term number

This formula allows you to calculate any term in the sequence directly without having to calculate all the previous terms.

Note: The common ratio (r) can be any real number except zero. If r = 1, the sequence becomes a constant sequence.

How to Use the Calculator

Enter the first term (a) of the geometric sequence
Enter the common ratio (r) of the sequence
Enter the term number (n) you want to calculate
Click the "Calculate" button
View the result and see the calculation details

The calculator will display the nth term of the geometric sequence using the explicit formula, along with a step-by-step explanation of the calculation.

Worked Examples

Example 1: Simple Geometric Sequence

Given a geometric sequence with first term a = 2 and common ratio r = 3, find the 5th term.

a₅ = 2 × 3^(5-1) = 2 × 3⁴ = 2 × 81 = 162

The 5th term of the sequence is 162.

Example 2: Fractional Common Ratio

Given a geometric sequence with first term a = 5 and common ratio r = 1/2, find the 4th term.

a₄ = 5 × (1/2)^(4-1) = 5 × (1/2)³ = 5 × 1/8 = 5/8

The 4th term of the sequence is 5/8.

Example 3: Negative Common Ratio

Given a geometric sequence with first term a = 10 and common ratio r = -2, find the 6th term.

a₆ = 10 × (-2)^(6-1) = 10 × (-2)⁵ = 10 × (-32) = -320

The 6th term of the sequence is -320.

Frequently Asked Questions

What is the difference between arithmetic and geometric sequences?

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 be negative?

Yes, the common ratio can be any real number except zero. A negative common ratio results in alternating signs in the sequence.

What happens if the common ratio is 1?

If the common ratio is 1, the sequence becomes a constant sequence where all terms are equal to the first term.

How do I find the common ratio if I know two terms?

If you know the first term (a) and the second term (b), you can find the common ratio by dividing the second term by the first term: r = b/a.

What are some real-world applications of geometric sequences?

Geometric sequences are used in finance for compound interest calculations, in physics for modeling exponential growth or decay, and in computer science for algorithms that involve recursive multiplication.