Detect arithmetic or geometric sequences and find the nth term and sum.
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How do you find the nth term and sum of a number sequence?
For an arithmetic sequence, aₙ = a₁ + (n − 1)d and Sₙ = n⁄2 (2a₁ + (n − 1)d). For a geometric sequence, aₙ = a₁ · r^(n−1) and Sₙ = a₁(rⁿ − 1) ⁄ (r − 1). Arithmetic adds a constant difference; geometric multiplies by a constant ratio. Example: for 2, 5, 8, 11 the difference is 3, so a₁₀ = 29 and the first ten terms sum to 155.
Understanding your result
A sequence is arithmetic when each term increases by the same amount (the common difference) and geometric when each term is multiplied by the same factor (the common ratio). If neither holds, no single nth-term formula exists and the tool says so rather than guessing.
Formula and method
Arithmetic: aₙ = a₁ + (n − 1)d and Sₙ = n⁄2 (2a₁ + (n − 1)d). Geometric: aₙ = a₁ · r⁽ⁿ⁻¹⁾ and Sₙ = a₁ (rⁿ − 1) ⁄ (r − 1).
Worked example
For 2, 5, 8, 11 the difference is 3, so a₁₀ = 2 + 9 × 3 = 29 and the first ten terms sum to 155.
How to use this tool
- Enter the known terms of your sequence.
- Optionally enter which term number you want.
- Read the type, formula, nth term and sum.
Common mistakes to avoid
- Entering numbers in the wrong order.
- Expecting a formula for a non-arithmetic, non-geometric list.
About the Number Sequence Calculator
The Number Sequence Calculator works out whether your numbers form an arithmetic sequence (constant difference) or a geometric sequence (constant ratio), then gives the common difference or ratio, the nth-term formula, any term you ask for, and the sum of the first n terms.
Who should use this tool
Students studying sequences and series, teachers preparing examples, and anyone spotting a pattern in a list of numbers.
Benefits
- Detects arithmetic or geometric patterns automatically.
- Builds the nth-term formula for you.
- Calculates any term and the running sum.
- Previews the next terms in the sequence.
Practical use cases
- Finding the 100th term of 2, 5, 8, 11, …
- Summing the first 20 terms of a geometric series.
- Checking homework on common difference and ratio.
Frequently asked questions
What is the difference between arithmetic and geometric?
In an arithmetic sequence you add the same number each step; in a geometric sequence you multiply by the same number each step.
Why does it say no pattern?
Your numbers have neither a constant difference nor a constant ratio, so they are not a simple arithmetic or geometric sequence — for example 1, 4, 9, 16 (squares).