Arithmetic sequences and their sums

An arithmetic sequence has a constant difference d between consecutive terms. Its nth term is uₙ = u₁ + (n − 1)d, and the sum of its first n terms is Sₙ = n[2u₁ + (n − 1)d]/2.

The idea

Find the common difference by subtracting one term from the next. The difference can be positive, negative or zero. There are n − 1 steps between the first term and the nth term. To sum a finite arithmetic sequence, multiply the number of terms by the mean of the first and last terms.

Worked example

For 5, 8, 11, 14, … find the tenth term and the sum of the first ten terms.

u₁ = 5 and d = 3.
u₁₀ = 5 + 9 × 3 = 32.
S₁₀ = 10 × (5 + 32)/2 = 185.

A mistake to avoid

Using nd instead of (n − 1)d shifts every answer by one step. Check your formula at n = 1.

Check your understanding

The first term is 20 and the common difference is −4. What is the sixth term?

Show the explanation

u₆ = 20 + 5 × (−4) = 0.

Explore the lesson

This guide connects to Arithmetic sequences and series in the IB Mathematics: Analysis and Approaches course. The first two published lessons per subject are free previews; further interactive lessons require an account.