Number Sequence Calculator
Arithmetic: each term adds a constant difference. Geometric: each term multiplies by a constant ratio.
Sequence type
nth term aₙ:
Sum of first n terms Sₙ:
Arithmetic vs. geometric sequences
In an arithmetic sequence, each term is found by adding a fixed common difference (d) to the previous one — the nth term is a₁ + (n−1)d, and the sum of the first n terms averages the first and last term and multiplies by the number of terms. In a geometric sequence, each term is found by multiplying the previous one by a fixed common ratio (r) — the nth term is a₁×r^(n−1), and the sum follows a compound-growth-style formula that differs when r equals exactly 1.
Frequently asked questions
How can I tell if a sequence is arithmetic or geometric? Check whether consecutive terms share a constant difference (arithmetic) or a constant ratio (geometric) — subtract consecutive terms to test for arithmetic, or divide consecutive terms to test for geometric.
What happens if the common ratio is negative? The sequence alternates in sign from term to term (e.g., 2, −6, 18, −54...) — the formulas still apply correctly, since they don't assume a positive ratio.
Where are these sequences used in real life? Arithmetic sequences describe steady, constant-increment patterns like simple interest or a fixed savings schedule, while geometric sequences describe compounding patterns like compound interest, population growth, or radioactive decay (in reverse).