$Arithmetic Sequence$

Arithmetic Sequence

The calculator finds the n-th term, the common difference and the sum of the first n terms of an arithmetic sequence from any known data and checks whether a number is a term. Step-by-step solution.

What is known:

First term and common difference

First and n-th term

Any two terms

Is a number a term of the sequence

First term a₁

Common difference d

Term number n

You can enter whole numbers, decimals and fractions: 2, −1.5, 1/3

Test yourself

The arithmetic sequence calculator finds the n-th term, the common difference and the sum of the first n terms. A sequence can be given by its first term and common difference, by the first and the n-th term, or by any two terms. The calculator also checks whether a number is a term of the sequence. You can enter whole numbers, decimals and fractions — the answer is exact, with a step-by-step solution and a table of the first terms.

What is an arithmetic sequence

An arithmetic sequence (arithmetic progression) is a sequence of numbers in which each term after the first is obtained by adding the same number d to the previous term. This number is called the common difference.

a_n+1 = a_n + d

For example, 2, 5, 8, 11, 14, … is an arithmetic sequence with the first term a₁ = 2 and the common difference d = 3. If d > 0 the sequence is increasing, if d < 0 it is decreasing, and if d = 0 all terms are equal.

Main formulas

What to find	Formula
n-th term	a_n = a₁ + d(n − 1)
Common difference	d = a_n+1 − a_n = (a_n − a₁) / (n − 1)
Sum of the first n terms	S_n = (a₁ + a_n) / 2 · n
Sum from a₁ and d	S_n = (2a₁ + d(n − 1)) / 2 · n
Mean property	a_n = (a_n−1 + a_n+1) / 2

Every term of an arithmetic sequence except the first is the arithmetic mean of its neighbors — this is where the name comes from.

Example 1. a₁ = 2, d = 3. Find a₁₀ and S₁₀.
a₁₀ = 2 + 3 · (10 − 1) = 29
S₁₀ = (2 + 29) / 2 · 10 = 155

Example 2. a₃ = 11, a₇ = 23. Find the common difference and the first term.
a₇ − a₃ = 4d ⇒ d = (23 − 11) / 4 = 3
a₁ = a₃ − 2d = 11 − 6 = 5

How to check whether a number is a term

A number x is a term of the sequence if there is a natural number n such that x = a₁ + d(n − 1). Solve for the position: n = (x − a₁) / d + 1. If n is a natural number, x is the n-th term; otherwise x is not in the sequence.

Example 3. Is 47 a term of 3, 7, 11, …?
n = (47 − 3) / 4 + 1 = 12 — yes, 47 = a₁₂.

Gauss's sum formula

The natural numbers 1, 2, 3, …, n form an arithmetic sequence with a₁ = 1 and d = 1, so their sum is n(n + 1) / 2. For example, 1 + 2 + … + 100 = 100 · 101 / 2 = 5050.