What is an arithmetic sequence?

An arithmetic sequence changes by the same common difference from one term to the next.

What is a geometric sequence?

A geometric sequence multiplies each term by the same common ratio.

Can a list have more than one rule?

Yes. A short custom list can fit many rules, so the calculator reports a likely pattern and explains the assumption.

How do I find the next term in a sequence?

Compare consecutive differences or ratios, identify the rule, and apply it to the last term. Three or more terms make a suggested pattern easier to verify.

Can one sequence have more than one rule?

Yes. A short list can fit several patterns, so treat automatic detection as a helpful suggestion. Add more terms or use the context of the problem to confirm the intended rule.

Can I share a calculation?

Yes. Use Share to copy a URL containing the current calculator values, then open that link on another device or send it to someone else.

Sequence Calculator – Arithmetic, Geometric & Fibonacci

A clear answer before the details

An arithmetic sequence adds a constant difference, while a geometric sequence multiplies by a constant ratio. Fibonacci sequences add the previous two terms.

Recognise arithmetic and geometric sequences

A sequence calculator looks for a rule connecting terms. An arithmetic sequence calculator checks for a constant common difference, such as 2, 5, 8, 11, where d = 3. A geometric sequence calculator checks for a constant common ratio, such as 3, 6, 12, 24, where r = 2. Once the rule is identified, the nth-term formula can find a distant term without listing every value.

Fibonacci and custom number sequences

A Fibonacci sequence calculator uses the familiar rule that each term is the sum of the previous two: 1, 1, 2, 3, 5, 8. A general number sequence calculator can also detect simple arithmetic or geometric patterns and report when a short custom list is ambiguous. Use at least three starting terms, then check the suggested difference, ratio, next term, and sum against the original values.

Formula guide

The key equations are separated below. Start with the calculator mode, then choose the shape or model that matches your problem.

Arithmetic sequence

aₙ = a₁ + (n − 1)d

d is the constant difference.

Geometric sequence

aₙ = a₁rⁿ⁻¹

r is the constant ratio.

Fibonacci sequence

Fₙ = Fₙ₋₁ + Fₙ₋₂

Each term adds the two previous terms.

Detect an arithmetic pattern

d = aₙ − aₙ₋₁

Compare adjacent terms; a constant difference indicates arithmetic growth.

Detect a geometric pattern

r = aₙ ÷ aₙ₋₁

Compare adjacent terms; a constant ratio indicates geometric growth.

Worked example

For 2, 5, 8, the common difference is 3, so the fourth term is 11 and the first four terms sum to 26.

Formulas for common use cases

Use the matching equation for the problem you are solving, then enter its variables in the calculator above.

Find the next term

next = last + d

Use this for an arithmetic pattern.

Repeated growth

next = last × r

Use this for a geometric pattern.

Sum arithmetic terms

Sₙ = n(a₁ + aₙ) ÷ 2

Pair the first and last terms for a quick sum.

How to use this calculator

Choose the sequence type or automatic detection.
Enter the first terms separated by commas.
Enter n when you need a specific term or sum.
Review the table and substitution steps.

Where it helps

Sequences appear in savings plans, repeated growth, patterns, algorithms, and algebra lessons. The term table makes the relationship visible, while the formula turns a pattern into a rule that can calculate a distant term without listing every value.

Common mistakes

A short custom sequence can have multiple plausible rules. Automatic detection works best with at least three terms and clear arithmetic or geometric behaviour. Check whether a ratio is undefined when a previous term is zero, and do not confuse a common difference with a common ratio.