Definition of an arithmetic sequence

What is an arithmetic sequence?

An arithmetic sequence is a list of numbers u_0, u_1, u_2, … such that, to move from one term to the next, the same number is always added. This number is called the common difference, denoted by r.

In other words, for any integer n: u(n+1) = u(n) + r

Practical example

Let’s take u_0 = 4 and r = 3. We get: u_0 = 4, u_1 = 7, u_2 = 10, u_3 = 13, ...

n	u_n
0	4
1	7
2	10
3	13

Each term is obtained by adding 3 to the previous one.

How do you recognise an arithmetic sequence?

We calculate the difference between two consecutive terms. If this difference is constant, the sequence is arithmetic. For example, for 2, 5, 8, 11: 5–2 = 3, 8–5 = 3, 11–8 = 3, so r = 3.

Pitfalls to avoid

• Do not confuse this with a geometric sequence, where you multiply by a common ratio instead of adding.
• The common difference r may be negative (a decreasing sequence) or zero (a constant sequence): these are indeed arithmetic sequences.
• Check the difference across several terms, not just the first two, before concluding that the sequence is arithmetic.