# Arithmetic Sequence Calculator

Calculate the nth term and sum of an arithmetic sequence. Enter the first term, common difference, and term number.

## What this calculates

Find any term and the cumulative sum of an arithmetic sequence. Enter the first term, common difference, and the desired term number to get instant results along with a sequence preview.

## Inputs

- **First Term (a₁)** — The first term of the arithmetic sequence.
- **Common Difference (d)** — The constant difference between consecutive terms.
- **Term Number (n)** — min 1 — Which term to calculate (n ≥ 1).

## Outputs

- **nth Term (aₙ)** — The value of the nth term: a₁ + (n-1)d.
- **Sum of First n Terms (Sₙ)** — The sum of all terms from a₁ to aₙ.
- **Sequence Preview** — formatted as text — The first 5 terms of the sequence.

## Details

An arithmetic sequence (or arithmetic progression) is a sequence of numbers where the difference between consecutive terms is constant. This constant is called the common difference (d). For example, the sequence 2, 5, 8, 11, 14 has a common difference of 3.

The nth term of an arithmetic sequence is given by aₙ = a₁ + (n-1)d, where a₁ is the first term and d is the common difference. The sum of the first n terms is Sₙ = n/2 × (2a₁ + (n-1)d), which can also be written as Sₙ = n/2 × (a₁ + aₙ), the average of the first and last term multiplied by the number of terms.

Arithmetic sequences appear throughout mathematics and everyday life: equally spaced payments, linear depreciation, stacking patterns, and evenly distributed points. The famous story of young Gauss summing the integers from 1 to 100 (getting 5050) uses the arithmetic series sum formula with a₁ = 1, d = 1, and n = 100.

## Frequently Asked Questions

**Q: What is an arithmetic sequence?**

A: An arithmetic sequence is a list of numbers where each term is obtained by adding a fixed value (the common difference) to the previous term. Examples: 3, 7, 11, 15 (d = 4) and 20, 15, 10, 5 (d = -5).

**Q: How do I find the nth term?**

A: Use the formula aₙ = a₁ + (n-1)d. For example, to find the 50th term of the sequence starting at 3 with d = 4: a₅₀ = 3 + (50-1) × 4 = 3 + 196 = 199.

**Q: How do I find the sum of the first n terms?**

A: Use Sₙ = n/2 × (a₁ + aₙ) or equivalently Sₙ = n/2 × (2a₁ + (n-1)d). For the first 100 positive integers: S₁₀₀ = 100/2 × (1 + 100) = 50 × 101 = 5050.

**Q: What is the difference between arithmetic and geometric sequences?**

A: In an arithmetic sequence, you add a constant (common difference) to get the next term. In a geometric sequence, you multiply by a constant (common ratio). Arithmetic: 2, 5, 8, 11. Geometric: 2, 6, 18, 54.

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Source: https://vastcalc.com/calculators/math/arithmetic-sequence
Category: Math
Last updated: 2026-04-21