# Arithmetic Progression Calculator — nth Term & Sum

> Find the nth term and the sum of an arithmetic progression from the first term, common difference and number of terms — with the formula and steps shown. Free.

- **Category:** Math
- **Interactive calculator:** https://youcalc.com/en/math/arithmetic-progression-calculator/
- **Price:** Free, no sign-up required

## Overview

An arithmetic progression (or arithmetic sequence) is a list of numbers in which each term is reached by adding the same fixed amount — the common difference d — to the one before it. Give this calculator the first term a₁, the common difference d and how many terms you want (n), and it returns the nth term aₙ, the sum of those n terms Sₙ, and a preview of the sequence itself. It works for increasing sequences (positive d), decreasing ones (negative d) and constant ones (d = 0), with whole or fractional terms.

## How to read your result

The large figure is the sum Sₙ — the total you get by adding the first n terms together. Beside it sits the nth term aₙ, which is the value of the last term in that run. The sequence preview lists the leading terms in order so you can see the pattern at a glance; when n is large the preview is capped, but aₙ still shows where the run ends. The worked-solution panel substitutes your numbers into the two formulas so you can check every step by hand.

## Method

In an arithmetic progression the nth term is aₙ = a₁ + (n − 1)·d, because you start at a₁ and add the common difference d a total of (n − 1) times to reach the nth position. The sum of the first n terms — an arithmetic series — is the number of terms multiplied by the average of the first and last term: Sₙ = n·(a₁ + aₙ) / 2. Substituting aₙ gives the equivalent closed form Sₙ = n/2·(2·a₁ + (n − 1)·d). The number of terms n must be a positive whole number; a₁ and d can be any real numbers, including negatives and fractions.

## Example

- **Setup:** First term a₁ = 2, common difference d = 3, and n = 10 terms.
- **Result:** The sequence is 2, 5, 8, 11, 14, 17, 20, 23, 26, 29. The 10th term is a₁₀ = 2 + (10 − 1)·3 = 29, and the sum of all ten terms is S₁₀ = 10·(2 + 29) / 2 = 155.

## Frequently asked questions

### What is the difference between the nth term and the sum?

The nth term aₙ is a single value — the number sitting in position n of the sequence — found with aₙ = a₁ + (n − 1)·d. The sum Sₙ adds up every term from the first through the nth: Sₙ = n·(a₁ + aₙ) / 2. For a₁ = 2, d = 3, n = 10 the 10th term is 29, while the sum of all ten terms is 155.

### How do I find the common difference?

The common difference d is the gap between any term and the one before it, so subtract a term from its successor: d = a₂ − a₁ = a₃ − a₂, and so on. If consecutive terms do not share a constant difference, the sequence is not arithmetic. A positive d gives an increasing sequence, a negative d a decreasing one, and d = 0 a constant sequence where every term equals a₁.

### Can the common difference or the terms be negative or fractional?

Yes. The first term a₁ and the common difference d can be any real numbers — negative, zero or fractional. With a₁ = 5 and d = −2 the sequence counts down: 5, 3, 1, −1, … . Only the number of terms n is restricted: it must be a whole number of 1 or more, since you cannot have a fractional count of terms.

### Why is the sum formula n·(a₁ + aₙ) / 2?

Pairing the first term with the last, the second with the second-to-last, and so on, every pair adds to the same total a₁ + aₙ. There are n/2 such pairs, so the sum is n·(a₁ + aₙ) / 2 — the trick attributed to Carl Friedrich Gauss, who is said to have summed 1 to 100 as 50 pairs of 101 to get 5050.

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## Sources

- https://mathworld.wolfram.com/ArithmeticSeries.html
- https://en.wikipedia.org/wiki/Arithmetic_progression

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