Arithmetic Sequences: Find the nth Term and the Sum

An arithmetic sequence goes up (or down) by the same amount each time. That amount is the common difference, d. In 3, 7, 11, 15, … the difference is 4.

Try it with your own problem ↓

The steps

Subtract any term from the one after it to get d. Check that it is the same each time.
The first term is a₁. The nth term is aₙ = a₁ + (n − 1) × d.
To find a particular term, put n in the formula.
The sum of the first n terms is Sₙ = n × (a₁ + aₙ) ÷ 2: the count times the average of the first and last terms.

A worked example

For 3, 7, 11, 15, …, find the 50th term and the sum of the first 50 terms.

Common difference: 7 − 3 = 4 (and 11 − 7 = 4, 15 − 11 = 4), so d = 4.
Formula: aₙ = 3 + (n − 1) × 4, which simplifies to 4n − 1.
50th term: a₅₀ = 3 + 49 × 4 = 3 + 196 = 199.
Sum: S₅₀ = 50 × (3 + 199) ÷ 2 = 50 × 101 = 5,050.

The 50th term is 199 and the sum of the first 50 terms is 5,050. The next terms after 15 are 19, 23, 27.

Try it with your own problem

Open the free The Number Staircase with this example already filled in. It shows every step, and you can type over it with your own numbers.

Work it out: 3, 7, 11, 15Opens The Number Staircase with this example

Or try these

Common mistakes

Using n instead of n − 1. The first term has no differences added, so it is (n − 1) steps from the start.
Mixing up terms and sums. aₙ is one term. Sₙ is the total of n terms.
Forgetting to halve in the sum. n × (a₁ + aₙ) ÷ 2 has the ÷ 2.
Calling a sequence arithmetic when it multiplies. 2, 6, 18, 54 has a common ratio, not a difference. That is geometric.

Questions people ask

What is the difference between arithmetic and geometric sequences?

Arithmetic sequences add the same amount each step. Geometric sequences multiply by the same amount.

How do I find which term equals 99?

Solve 4n − 1 = 99: 4n = 100, so n = 25.

Where does the sum formula come from?

Pair the first and last terms, the second and second-to-last, and so on. Each pair adds to the same total, a₁ + aₙ, and there are n ÷ 2 pairs.

Keep learning

Show The Steps is free, with no accounts or ads, and everything is worked out in your own browser. Start at the home page.