By Chris Deziel • Updated Aug 30, 2022

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An arithmetic sequence is a list of numbers arranged in order, where each term differs from the previous by a fixed amount. For example, the sequence 3, 6, 9, 12, … increases by a constant difference of 3. In contrast, the geometric sequence 1, 3, 9, 27, 81, … multiplies each term by 3, so it is not arithmetic.

While short sequences can be identified visually, long sequences—thousands of terms—require a systematic approach. The arithmetic‑sequence formula lets you jump directly to any term without writing the entire list.

Deriving the Arithmetic‑Sequence Formula

Let a denote the first term and d the common difference. The sequence can be written as:

a, a + d, a + 2d, a + 3d, …

For the nth term, the general formula is:

x_n = a + d (n – 1)

Example: Find the 10th term of the sequence 3, 6, 9, 12, ….

x₁₀ = 3 + 3 (10 – 1) = 30

Listing the terms confirms the result.

Sample Problem: Building a Rule from a Sequence

Often a problem presents a numeric list and asks you to write a formula that generates any term. Consider the sequence:

7, 12, 17, 22, 27, …

Here, a = 7 and d = 5. Plugging into the formula gives:

x_n = 7 + 5(n – 1) = 2 + 5n

With this rule, you can find any term or identify which position a given number occupies.

• 100th term: n = 100 → x₁₀₀ = 2 + 5·100 = 502

• Which term is 377? Solve for n:

n = (x_n – 2)/5 = (377 – 2)/5 = 75

Thus, 377 is the 75th term.

Mastering this formula enables you to solve arithmetic‑sequence problems efficiently, no matter how many terms the sequence contains.