What is an arithmetic progression (AP)?

An arithmetic progression is a sequence of numbers in which each term after the first is formed by adding a fixed constant (the common difference d) to the preceding term.

What is a geometric progression (GP)?

A geometric progression is a sequence where each term after the first is found by multiplying the preceding term by a fixed non-zero constant (the common ratio r).

Can an infinite geometric series be summed?

Yes, but only if the absolute value of the common ratio is strictly less than 1 (|r| < 1). In that case, the infinite sum converges to S∞ = a₁ / (1 - r).

Math

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Number Sequence Calculator — Arithmetic, Geometric & Fibonacci

Free number sequence calculator to find the nth term, partial sums, and complete term lists for arithmetic, geometric, and Fibonacci sequences.

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How to use this calculator

👉 Fill in the boxes below and your answer appears instantly — no maths needed, we do it all for you! 🎉

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In plain English — what does this do?

🔢 This tool does the maths for you! Just type in your numbers, and it gives you the answer right away. No need to count on your fingers or use a pen and paper.

📌 Direct Answer & Summary

The Number Sequence Calculator determines any nth term, sequence sums, and generates full term lists for arithmetic progressions (AP), geometric progressions (GP), and Fibonacci recurrence sequences.

First Term (a₁)

Common Difference (d)

Target Term Index (n)

What is Number Sequence Calculator — Arithmetic, Geometric & Fibonacci?

The Number Sequence Calculator determines any nth term, sequence sums, and generates full term lists for arithmetic progressions (AP), geometric progressions (GP), and Fibonacci recurrence sequences.

How to use it

1️⃣ Select sequence type: Arithmetic, Geometric, or Fibonacci.
2️⃣ Enter the initial term (a₁), common difference (d) or common ratio (r), and target term index (n).
3️⃣ View the exact nth term (aₙ) and the series sum (Sₙ).
4️⃣ Inspect the generated list of terms and the step-by-step algebraic formula.

Formula

Arithmetic: aₙ = a₁ + (n - 1)d; Sum Sₙ = (n / 2)(a₁ + aₙ). Geometric: aₙ = a₁ × rⁿ⁻¹; Sum Sₙ = a₁(1 - rⁿ) / (1 - r). Fibonacci: Fₙ = Fₙ₋₁ + Fₙ₋₂.

💡 See it in action — a real example

For an arithmetic sequence with a₁ = 2, difference d = 5, and n = 20: 20th term a₂₀ = 2 + (20 - 1) × 5 = 97. Sum of first 20 terms S₂₀ = (20 / 2) × (2 + 97) = 990.

❓ Common questions

What is an arithmetic progression (AP)?: An arithmetic progression is a sequence of numbers in which each term after the first is formed by adding a fixed constant (the common difference d) to the preceding term.
What is a geometric progression (GP)?: A geometric progression is a sequence where each term after the first is found by multiplying the preceding term by a fixed non-zero constant (the common ratio r).
Can an infinite geometric series be summed?: Yes, but only if the absolute value of the common ratio is strictly less than 1 (|r| < 1). In that case, the infinite sum converges to S∞ = a₁ / (1 - r).