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In an arithmetic sequence each term increases by a fixed amount called the common difference d. The nth term is the first term plus (n − 1) steps of d: aₙ = a₁ + (n − 1)d. The sum of the first n terms is the number of terms times the average of the first and last: Sₙ = n(a₁ + aₙ)/2. Enter the first term, common difference and term number. For a₁ = 2 and d = 3, the 10th term is 29 and the first 10 terms add to 155.

Arithmetic Sequence Calculator — nth term and sum

First term 2, common difference 3, 10 terms.

nth term29
Sum of first n terms
155

Quick examples

How it's calculated

  1. nth term = first term + (n − 1) × common differencea10=2+(10−1)×3a_{10} = 2 + (10 - 1) \times 3
    firstTerm
    = 2
    commonDifference
    = 3
    n
    = 10
    29
nth term29

How it works

An arithmetic sequence is a list of numbers where each term is the previous one plus a constant, the common difference d. Starting from the first term a₁, the sequence is a₁, a₁ + d, a₁ + 2d, and so on.

To reach the nth term you take n − 1 steps of size d from the first term, per Wolfram MathWorld:

aₙ = a₁ + (n − 1)d

The sum of the first n terms has a neat shortcut. Pairing the first and last terms, the second and second-to-last, and so on, every pair adds to a₁ + aₙ, which gives:

Sₙ = n(a₁ + aₙ) / 2

This is the number of terms times the average of the first and last term. It is the same idea Gauss famously used to add 1 through 100 in seconds.

Worked example

Take a₁ = 2 and d = 3. The 10th term is 2 + (10 − 1) × 3 = 2 + 27 = 29. The sum of the first 10 terms is 10 × (2 + 29) / 2 = 10 × 31 / 2 = 155. The sequence is 2, 5, 8, 11, …, 29.

Frequently asked questions

What is an arithmetic sequence?

It is a sequence where each term differs from the one before it by the same fixed amount, the common difference. For example 2, 5, 8, 11, … has a common difference of 3.

How do you find the nth term?

Add n − 1 copies of the common difference to the first term: aₙ = a₁ + (n − 1)d. For a₁ = 2 and d = 3, the 10th term is 2 + 9 × 3 = 29.

How do you find the sum of an arithmetic sequence?

Multiply the number of terms by the average of the first and last term: Sₙ = n(a₁ + aₙ)/2. For the first 10 terms of 2, 5, 8, …, that is 10 × (2 + 29)/2 = 155.

What is the common difference?

It is the constant gap between consecutive terms. Subtract any term from the next to find it — in 3, 7, 11, 15 the common difference is 4. A negative common difference makes a decreasing sequence.

Can the common difference be negative or a decimal?

Yes. A negative difference gives a decreasing sequence (20, 16, 12, …), and a decimal difference gives fractional steps (0.5, 1, 1.5, …). The formulas work the same way.

How is this different from a geometric sequence?

An arithmetic sequence adds a constant each step, while a geometric sequence multiplies by a constant ratio. So 2, 5, 8 is arithmetic (add 3), but 2, 6, 18 is geometric (multiply by 3).

How we know this is right

Last reviewed
Aug 8, 2026
Precision
Rounded to 4 decimal places.
Read our methodology

Sources