Sequence and Series Question Generator

What this generator does

Draws a first term and a common difference (arithmetic) or common ratio (geometric), then answers the chosen question — the nth term, or the sum of the first n terms — using the standard closed-form formula. That answer is then independently recomputed by literally building the sequence one term at a time (and running total, for a sum), and the question is only returned once the two agree exactly.

How to use this tool

1. Choose arithmetic or geometric, nth term or sum, a level, and how many questions you want.
2. Work each one out using the nth-term or sum formula for that sequence type.
3. Reveal the answers to check your own, or the working to see the formula applied.
4. Seed the set if two people should get the identical questions.

Understanding the controls

Question type

Arithmetic or geometric, nth-term or sum-of-terms — four distinct question shapes from one page.

Level

Gentle stays positive with a short sequence; standard allows negative terms and ratios; stretch widens both the numbers and how many terms are involved.

How many questions

Between 1 and 20 — ten is a reasonable page once the working is shown for each one.

Seed (optional)

Any word reproduces the identical set of questions, which is how you give two people the same sheet.

Common use cases

• Practice at the nth-term formula for arithmetic and geometric sequences
• Practice at the sum formulas for arithmetic and geometric series
• A worksheet that never runs out of fresh questions of a chosen type
• Checking a formula, a spreadsheet or your own working against a known-correct answer
• Giving a class the identical sequences so answers can be compared term by term, not just at the end

How this generator works

A geometric sum in particular is exactly the kind of formula that looks fine and occasionally is not: a*(r^n - 1)/(r - 1) can only ever be trusted if that division comes out even, so it is computed with exact integer (BigInt) arithmetic and the division is checked to be exact rather than assumed. But the real check is independent of the formula altogether — every answer, arithmetic or geometric, nth-term or sum, is also built by starting at the first term and repeatedly adding the difference or multiplying by the ratio, term by term, keeping a running total for a sum question. A question is only ever shown once the formula's answer and the built-up answer are the same number. A common difference of zero and a common ratio of 0, 1 or -1 are never drawn, because each collapses the sequence into something that is not a useful question (a constant run, or a trivial alternating sign).

Randomness and fairness

The first term, the common difference or ratio, and which term or how many terms all come from your browser's cryptographic random source. Giving a seed replaces that source with a repeatable one, so the same set of questions can be reproduced; the seed is for reproducibility, not cryptographic strength.

For how randomness is produced across the whole site, see how Generate Random works.

Limitations and good to know

• Every sequence here has a whole-number first term, difference or ratio, and answer — irrational or fractional sequences are not covered.
• A common ratio is never 0, 1 or -1, so the trivial and alternating-sign edge cases of a geometric sequence are not exercised here.
• Nothing is marked — the answer and working are revealed for you to compare against your own, not entered and checked.
• A generated set is not saved anywhere, so seed anything you intend to print or reissue.

Privacy and your data

Every question is built and checked in your browser. Your seed, level and the questions themselves are never transmitted or retained.