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Algebra

Nth Term Calculator

Enter a number sequence to identify whether it is arithmetic, geometric or quadratic — then derive the nth term formula with full step-by-step working and look up any specific term.

Nth term calculator

Separate terms with commas or spaces. Enter at least 3–4 terms for detection.

What is the th term?
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Step by step calculations

Enter a number sequence and press Calculate to see the step by step formula derivation.

Nth Term Calculator Explained

Core concepts, classroom applications, and plain-English explanations for all audiences.

When would you actually use this?

Imagine chairs are arranged in rows at a concert: row 1 has 3 chairs, row 2 has 6, row 3 has 9. How many chairs are in row 20? Instead of counting all the way, the nth term rule gives you the answer directly: 3n, so row 20 has 60 chairs.

Worked example

"A trainee starts on £22,000 and gets a £1,500 pay rise each year. What will they earn in year 8?"

Sequence: 22000, 23500, 25000, ...

Common difference: 1500

nth term: 1500n + 20500

Year 8: 1500(8) + 20500 = £32,500

Always check term 1: 1500(1) + 20500 = 22000 ✓

Common mistakes

  • ✗ Using the wrong starting point: the constant is the "zero-th term" (what comes before term 1), not the first term itself.
  • ✗ Wrong common difference: always subtract consecutive terms in the same order to check it is constant.

Quick test

Substitute n = 1 and n = 2 back into your formula. Both answers should match the original sequence.

Good to know

Three types of sequence

+d
Arithmetic sequences
An arithmetic sequence has a constant difference between consecutive terms. You add (or subtract) the same amount each time. The nth term formula is always linear: dn + c, where d is the common difference. These are the most common sequence type at secondary school.
3, 7, 11, 15, 19...
diff: +4, +4, +4, +4
nth term = 4n − 1

Check: 4(1)−1=3 ✓ 4(2)−1=7 ✓
×r
Geometric sequences
A geometric sequence has a constant ratio between consecutive terms. The nth term formula is exponential: a × r^(n−1). If r > 1 the sequence grows; if 0 < r < 1 it shrinks toward zero; if r is negative, terms alternate in sign.
2, 6, 18, 54, 162...
ratio: ×3, ×3, ×3, ×3
nth term = 2 × 3^(n−1)

Check: 2×3⁰=2 ✓ 2×3¹=6 ✓
n²
Quadratic sequences
A quadratic sequence has constant second differences. The nth term is an² + bn + c. Find a from the second difference (a = 2nd diff ÷ 2), then work back to b and c. This method appears in higher secondary school and GCSE/A-level maths.
1, 4, 9, 16, 25...
1st diff: 3, 5, 7, 9
2nd diff: 2, 2, 2 (constant!)
a=1, b=0, c=0
nth term = n²