Sequences & Series

Arithmetic Sequences

Writing explicit and recursive formulas for an arithmetic sequence.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

A Function Defined Only on Whole Numbers

5, 8, 11, 14, ... Before reading on, predict the next two terms, and think about how you'd describe a rule for finding the 100th term directly, without listing all 99 terms before it.

Definition — Arithmetic Sequence

A sequence where each term differs from the previous one by a constant amount, called the common difference, d. The nth term has an explicit formula aₙ = a₁ + (n − 1)d, and a recursive formula aₙ = aₙ₋₁ + d.

The explicit formula's structure isn't arbitrary — it's exactly the same structure as a linear function's slope-intercept form, y = mx + b, just restricted to whole-number inputs. Starting at a₁ and adding d a total of (n − 1) times to reach the nth term is precisely what 'slope times steps, plus a starting value' means — an arithmetic sequence is genuinely a linear function evaluated only at n = 1, 2, 3, ....

Worked Example — Finding an Explicit Formula

Find an explicit formula for 5, 8, 11, 14, .... The first term is a₁ = 5, and the common difference is d = 3 (each term is 3 more than the last). The explicit formula: aₙ = 5 + (n − 1)(3) = 3n + 2.

Worked Example — Finding a Specific Term

Using aₙ = 3n + 2, find the 40th term. a₄₀ = 3(40) + 2 = 122.

Worked Example — Writing a Recursive Formula

Write a recursive formula for the same sequence, 5, 8, 11, 14, .... State the starting term and the rule for the next term: a₁ = 5, and aₙ = aₙ₋₁ + 3 for n > 1.

Tip

The common difference is the same number that would be an arithmetic sequence's 'slope' if it were graphed as (n, aₙ) points — a fast way to double-check a found common difference is to confirm it matches consecutive term differences throughout the sequence, not just the first pair.

Common Mistakes

Using n instead of (n − 1) in the explicit formula, such as writing aₙ = a₁ + nd.
The first term, a₁, already accounts for zero additions of d — by the time n = 1, d has been added (1 − 1) = 0 times, which is why the formula uses (n − 1), not n.
Confusing a recursive formula (which needs a previous term to compute the next) with an explicit formula (which computes any term directly from n).
An explicit formula finds any term directly from its position n alone; a recursive formula requires already knowing the term before it — pick the right one for what a problem actually needs.

Key Takeaways

An arithmetic sequence adds the same common difference at every step.
Its explicit formula, aₙ = a₁ + (n − 1)d, mirrors a linear function's structure, restricted to whole-number inputs.
A recursive formula defines each term using the one before it, rather than directly from its position.

Summary

Arithmetic sequences are linear functions living on whole numbers. The next lesson looks at sequences that grow multiplicatively instead — the discrete cousin of exponential functions.

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