# Number Sequence Calculator

> Find the nth term and the sum of the first n terms of an arithmetic, geometric or Fibonacci sequence, plus the infinite sum where it exists.

Category: Mathematics · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/number-sequence-calculator

Enter how a sequence starts and how it grows, and get any term, the running total of the first n terms and the first ten terms listed.

Also called: Arithmetic sequence calculator, Geometric series calculator, nth term calculator.

## Inputs
- Sequence type (options: Arithmetic (add the same amount each time), Geometric (multiply by the same ratio), Fibonacci (1, 1, 2, 3, 5, …); default arithmetic)
- First term (a₁) (default 2)
- Common difference (d) (default 3)
- Common ratio (r) (default 2)
- Term number (n) (default 10)

## Results
- nth term
- Sum of the first n terms
- First terms
- Sum to infinity

## Formula

```
Arithmetic: aₙ = a₁ + (n − 1)d,  Sₙ = n(2a₁ + (n − 1)d) ÷ 2;  Geometric: aₙ = a₁·rⁿ⁻¹,  Sₙ = a₁(1 − rⁿ) ÷ (1 − r),  S∞ = a₁ ÷ (1 − r) for |r| < 1;  Fibonacci: Fₙ = Fₙ₋₁ + Fₙ₋₂
```

Where:
- a₁: first term
- d: common difference
- r: common ratio
- n: position of the term

## Worked example
1. a₁₀ = 2 + (10 − 1) × 3 = 2 + 27 = 29.
2. S₁₀ = 10 × (2 × 2 + 9 × 3) ÷ 2 = 10 × 31 ÷ 2 = 155.
3. Check: the first and last terms average (2 + 29) ÷ 2 = 15.5, and 15.5 × 10 = 155.

**Result:** In 2, 5, 8, …, the 10th term is 29 and the first ten terms add up to 155.

## What the result means

An arithmetic sequence grows by a fixed amount (like simple interest); a geometric one grows by a fixed factor (like compound interest), so it eventually outruns any arithmetic sequence when r > 1.

## Assumptions and limits
- The first term is term number 1. Fibonacci starts F₁ = F₂ = 1.
- Results are shown to 15 significant digits; very large geometric terms are approximate beyond that.

## Questions

### When does a geometric series have a finite sum to infinity?

Only when the ratio is between −1 and 1. Then the terms shrink fast enough that the total approaches a₁ ÷ (1 − r); for example 1 + ½ + ¼ + … = 2.

### How do I find the common difference?

Subtract any term from the next one. If the differences are not constant but the ratios are, the sequence is geometric.

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Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/number-sequence-calculator