Mathematics · Algebra
Number Sequence Calculator — Arithmetic, Geometric & Fibonacci
Find the n-th term, the sum of the first n terms and the first terms of an arithmetic progression (AP), geometric progression (GP) or Fibonacci-type sequence. Or type a few terms and let it spot whether they're arithmetic or geometric and continue them. Includes the sum to infinity of a converging GP.
Your inputs
Quick start — try a typical case
Find the n-th term and the sum of the first n terms.
Your result Live
Term 10 (a₁₀)
39
aₙ = 3 + (n − 1) × 4
- Sum of the First 10 Terms
- 210
- First 10 Terms
- 3, 7, 11, 15, 19, 23, 27, 31, 35, 39
How this is calculated
AP: aₙ = a₁ + (n − 1)d, Sₙ = n/2 × (2a₁ + (n − 1)d) · GP: aₙ = a₁rⁿ⁻¹, Sₙ = a₁(1 − rⁿ)/(1 − r), S∞ = a₁/(1 − r) for |r| < 1
- d, r
- Common difference (AP) and common ratio (GP)
An arithmetic sequence adds the same number each time (3, 7, 11, …), so the n-th term is the first plus n − 1 differences. A geometric sequence multiplies by the same ratio (2, 6, 18, …). Adding up the terms has neat formulas for both — Gauss's trick for arithmetic sums — and a geometric series with a ratio between −1 and 1 adds up to a finite total even with infinitely many terms.
Assumptions & sources
- Fibonacci-type terms are added exactly up to about 15 digits; very large terms are approximate.
Calculator ID MATH-SEQUENCE-001 · version 1.0.0
Common questions
What's the 10th term of 3, 7, 11, …?
a₁₀ = 3 + 9 × 4 = 39; the sum of the first 10 terms is 5 × (6 + 36) = 210.
What's the sum of 1 + 1/2 + 1/4 + … forever?
2 — a geometric series with r = 1/2 sums to a₁ ÷ (1 − r) = 1 ÷ 0.5.
What's the 20th Fibonacci number?
6,765 (starting 1, 1).