Sequence Calculator
Calculate the nth term and the sum of the first n terms of an arithmetic or geometric sequence.
Arithmetic vs. geometric sequences
An arithmetic sequence adds the same fixed amount (the common difference) to get each next term — like 3, 7, 11, 15. A geometric sequence multiplies by the same fixed amount (the common ratio) instead — like 3, 6, 12, 24. Recognizing which pattern applies is the first step to solving any sequence problem.
Geometric sequences grow (or shrink) much faster than arithmetic ones when the ratio is greater than 1, which is why compound interest, population growth, and radioactive decay are all modeled as geometric rather than arithmetic sequences.
Where sequences show up practically
Arithmetic sequences describe steady, linear patterns — evenly spaced seating rows, a savings plan with fixed regular deposits (before interest), or a countdown with equal intervals. Geometric sequences describe compounding or exponential patterns — investment growth, viral spread modeling, or the doubling pattern in some biological processes.
The sum formulas this calculator uses are especially useful for geometric sequences, where manually adding many terms would be impractical — the formula collapses that sum into a single calculation.
Frequently asked questions
How can I tell if a sequence is arithmetic or geometric?
Check whether consecutive terms differ by a constant amount (arithmetic, e.g., always +4) or a constant ratio (geometric, e.g., always ×3) — dividing consecutive terms will confirm geometric, subtracting will confirm arithmetic.
What happens to a geometric sequence if the ratio is between 0 and 1?
The sequence decreases toward zero rather than growing, since each term is a fraction of the previous one — this pattern describes things like radioactive decay or a bouncing ball losing height with each bounce.