Number Sequence Calculator
Generate arithmetic and geometric number sequences, calculate nth terms, step differences, and sequence sums.
Sequence Results
Adding versus multiplying
Sequences are classified by what takes you from one term to the next. An arithmeticsequence adds a constant difference: 3, 7, 11, 15 with d = 4, and the nth term isa₁ + (n−1)d. A geometric sequence multiplies by a constant ratio: 3, 6, 12, 24 with r = 2, and the nth term is a₁ × r^(n−1).
The difference in growth is the important part. Arithmetic sequences grow in a straight line; geometric ones grow exponentially, which is why compound interest, population models and viral spread are all geometric. Sums differ correspondingly: an arithmetic series isn/2 × (first + last) — the trick Gauss is said to have found as a schoolboy — while a geometric series is a₁(1 − rⁿ)/(1 − r).
An infinite geometric series converges only when |r| < 1, summing toa₁/(1 − r); otherwise it grows without bound. Not every sequence is one of these two — Fibonacci is defined recursively from the previous two terms, and squares and triangular numbers follow their own rules. A finite run of terms never proves a pattern; infinitely many rules fit any finite list.
Frequently asked questions
What is an arithmetic sequence?
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.
What is a geometric sequence?
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed common ratio.
What is the Fibonacci sequence?
Each term is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13. The ratio between consecutive terms converges on the golden ratio, roughly 1.618, and the sequence appears in phyllotaxis, branching patterns and a great deal of recreational mathematics.
Can I generate long sequences?
Yes. Enter any first term, common difference or ratio, and number of terms to generate sequences of any length.