Quick answer

The nth term equals the first term plus (n - 1) times the common difference: aₙ = a₁ + (n - 1)d.

Formula

  • aₙ = a₁ + (n - 1)d
  • aₖ = a₁ + (k - 1)d (position k)

Introduction

The arithmetic sequence formula is the fastest route to a distant term without writing every value in between. Once you trust the substitution order, most exercises become routine algebra.

If you are new to the vocabulary, start with what an arithmetic sequence is and return here for notation details.

On homework, write the formula before you plug in numbers. That habit makes errors easier to spot when a sign flips or when n is misread.

Formula parts and variables

a₁ is the first term: the anchor value at position 1. Every other term is measured relative to that start.

d is the common difference: the fixed change between neighbors. It may be negative, which creates a decreasing list.

n (or k) is the index of the term you want. It counts position, not the number of additions unless the problem defines otherwise.

(n - 1) is the number of d steps after a₁. There is one fewer gap than the index label because the first term already occupies position 1.

General nth term and equivalents

  • aₙ = a₁ + (n - 1)d
  • Equivalent: aₖ = a₁ + (k - 1)d for position k
  • Solve for d: d = (aₙ - a₁) / (n - 1) when n > 1

All three forms describe the same linear relationship. Choose the form that matches what the question asks you to find.

After substitution practice, move to how to find arithmetic sequence terms for a full listing workflow with verification steps.

Worked numbers with positive, negative, and fractional values of d appear in the examples article later in this blog cluster.

Step-by-step guide

  1. Write known values. Copy a₁, d, and the target index from the prompt.
  2. Substitute into the formula. Replace symbols before you simplify arithmetic.
  3. Simplify carefully. Multiply (n - 1) by d before adding to a₁.
  4. Check with a neighbor. If possible, confirm your term fits the constant difference rule.
  5. Optional calculator check. Enter the same a₁, d, and n in the calculator to compare lists.

Worked example

Find the 10th term when a₁ = 5 and d = 4.

Substitute: a₁₀ = 5 + (10 - 1)×4 = 5 + 36 = 41.

Sanity check: the list begins 5, 9, 13, … and each jump is 4, so reaching 41 on the 10th term is reasonable.