Arithmetic sequences
An arithmetic sequence adds the same fixed amount — the common difference — to get from one term to the next.
nth Term: an = a1 + d(n − 1) Sum of first n terms: Sn = n/2 × (2a1 + (n − 1)d) First term 2, common difference 3, 5th term a5 = 2 + 3 × (5 − 1) = 14 Sum of first 5 terms = 40 Geometric sequences
A geometric sequence multiplies by the same fixed factor — the common ratio — to get from one term to the next, so it grows (or shrinks) much faster than an arithmetic sequence with a similar-looking difference.
nth Term: an = a × rn−1 Sum of first n terms: Sn = a(1 − rn) ÷ (1 − r), or a × n when r = 1 First term 2, common ratio 3, 5th term a5 = 2 × 34 = 162 Sum of first 5 terms = 242 Sequences vs. permutations and combinations
Number sequences follow arithmetic, geometric, recursive, or other rule-based patterns to calculate ordered values over step indices. If you need to determine how many different arrangements or selections are possible from a set rather than continue a numerical pattern, use the Permutation & Combination Calculator.
Frequently asked questions
What's the difference between arithmetic and geometric sequences?
An arithmetic sequence adds a fixed amount each step (2, 5, 8, 11…), so it grows at a constant rate. A geometric sequence multiplies by a fixed factor each step (2, 6, 18, 54…), so it grows — or shrinks — much faster, and can flip sign every term if the ratio is negative.
Can the common ratio be negative, zero, or a fraction?
Yes to all three. A negative ratio makes the sequence alternate between positive and negative terms. A ratio between −1 and 1 makes the sequence shrink toward zero. A ratio of exactly 0 makes every term after the first equal to zero.
Why does the sequence list stop before term n?
The list shows the first 20 terms so it stays readable for large values of n — but the term value and sum shown above it are always calculated for the exact n you entered, not just the visible portion of the list.