nth Term Finder – Find Any Term of a Sequence Online

Find the nth term of any arithmetic or geometric sequence with our free online nth term finder. Enter the first two terms to calculate any specific term instantly with step-by-step solutions.

Sequence Type:

Examples:

First Term (a1):

Second Term (a2):

Find Term Number (n):

Understanding Sequences and nth Terms

A sequence is an ordered list of numbers following a pattern. The nth term formula lets you find any term in the sequence without calculating all the previous terms. This is essential for finding distant terms like the 100th or 1000th term efficiently.

There are two main types of sequences: arithmetic (adding a constant each time) and geometric (multiplying by a constant each time). Each has its own nth term formula that we'll explore below.

Sequence Formulas

Arithmetic Sequence

an = a1 + (n-1)d

where d = common difference = a2 - a1

Example: 3, 7, 11, 15... (d = 4)

Geometric Sequence

an = a1 x r^(n-1)

where r = common ratio = a2 / a1

Example: 2, 6, 18, 54... (r = 3)

Worked Examples

Example 1: Arithmetic Sequence

Sequence: 3, 7, 11, 15... Find the 25th term

a1 = 3, d = 7-3 = 4

a25 = 3 + (25-1)(4)

a25 = 3 + 96 = 99

Example 2: Decreasing Arithmetic

Sequence: 50, 47, 44... Find the 20th term

a1 = 50, d = 47-50 = -3

a20 = 50 + (20-1)(-3)

a20 = 50 - 57 = -7

Example 3: Geometric Sequence

Sequence: 2, 6, 18, 54... Find the 10th term

a1 = 2, r = 6/2 = 3

a10 = 2 x 3^(10-1)

a10 = 2 x 3^9 = 2 x 19683 = 39,366

Example 4: Geometric Decay

Sequence: 100, 50, 25... Find the 6th term

a1 = 100, r = 50/100 = 0.5

a6 = 100 x (0.5)^(6-1)

a6 = 100 x 0.03125 = 3.125

Quick Fact

The Fibonacci sequence (1, 1, 2, 3, 5, 8...) is neither arithmetic nor geometric – each term is the sum of the two previous terms. It has its own special nth term formula called Binet's formula, which involves the golden ratio φ = (1 + √5)/2.

Frequently Asked Questions

How do I know if a sequence is arithmetic or geometric?

Check the differences between consecutive terms. If they're constant, it's arithmetic. If the ratios between consecutive terms are constant, it's geometric. If neither, it might be a different type of sequence.

Can the common difference or ratio be negative?

Yes! A negative common difference creates a decreasing arithmetic sequence. A negative common ratio creates an alternating sequence (positive, negative, positive...). A ratio between -1 and 1 causes the terms to approach zero.

What if the first term is 0 in a geometric sequence?

If a1 = 0, every term in a geometric sequence is 0 (since you keep multiplying by r). This is a trivial case. Also, you can't find r by dividing a2/a1 if a1 = 0.

How do I find which term equals a specific value?

Set the nth term formula equal to your target value and solve for n. For arithmetic: n = (target - a1)/d + 1. For geometric: n = log_r(target/a1) + 1.

What are real-world applications of sequences?

Arithmetic sequences model linear growth (salary increases, depreciation). Geometric sequences model exponential growth/decay (population, compound interest, radioactive decay). Both appear in computer science algorithms and financial calculations.

Can n be a decimal or negative?

In standard sequence notation, n represents a position and must be a positive integer (1, 2, 3...). However, the formulas can be extended to real values for interpolation or continuous models.

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