Sequence Calculator
Find the nth term and the sum of an arithmetic or geometric sequence – or type a few terms and let the calculator recognise the pattern.
Arithmetic sequences
Each term is the previous one plus a fixed difference d: 5, 8, 11, 14, … has d = 3.
- nth term: aₙ = a₁ + (n − 1) · d
- Sum of the first n terms: Sₙ = n · (a₁ + aₙ) / 2
For 5, 8, 11, …: a₁₀ = 5 + 9 · 3 = 32 and S₁₀ = 10 · (5 + 32) / 2 = 185.
Geometric sequences
Each term is the previous one times a fixed ratio r: 3, 6, 12, 24, … has r = 2.
- nth term: aₙ = a₁ · rⁿ⁻¹
- Sum of the first n terms: Sₙ = a₁ · (1 − rⁿ) / (1 − r)
- If −1 < r < 1, the sum to infinity is a₁ / (1 − r).
For 3, 6, 12, …: a₁₀ = 3 · 2⁹ = 1,536 and S₁₀ = 3 · (2¹⁰ − 1) = 3,069. For 1, ½, ¼, … the infinite sum is 2.
Recognising a sequence
Choose From terms and type at least three terms. If the differences between them are equal, the sequence is arithmetic; if the ratios are equal, it is geometric. Anything else – like 1, 4, 9, 16 – is neither, and the calculator says so.
Where they appear
Arithmetic sequences describe steady growth: a savings plan with the same deposit every month, or seats in rows that each have 2 more. Geometric sequences describe percentage growth: compound interest, population growth or the halving of a radioactive sample.
Frequently asked questions
What is the 10th term of 5, 8, 11, …?
32: a₁₀ = 5 + 9 × 3.
How can I tell arithmetic from geometric?
Subtract neighbouring terms – if the result is always the same, it is arithmetic. Divide them – if the result is always the same, it is geometric.
When does a geometric series have a finite infinite sum?
When the ratio is between −1 and 1. Then the terms shrink towards 0 and the sum approaches a₁ / (1 − r).