Arithmetic progressions

Definition

A sequence (u_n) is arithmetic if each term is obtained by adding the same number – called the common difference and denoted by r – to the previous term: u_(n+1) = u_n + r for all integers n

Explicit formula

Given the first term u_0 and the common difference r, we can show that: u_n = u_0 + n * r

Example: u_0 = 5 and r = 3. Then u_n = 5 + 3n, so u_10 = 5 + 3*10 = 35.

If the sequence starts at a different index, for example u_1, the formula is adjusted as follows: u_n = u_1 + (n-1) * r.

Direction of variation

• If r > 0, the sequence is strictly increasing (it increases).
• If r < 0, the sequence is strictly decreasing (it decreases).
• If r = 0, the sequence is constant.

Sum of consecutive terms

To add up consecutive terms of an arithmetic sequence, we use: Sum = (number of terms) * (first term + last term) / 2

Example: the sum of the integers from 1 to 10 (an arithmetic sequence with common difference 1) is 10 * (1 + 10) / 2 = 55.

Common pitfall

Do not confuse the common difference r (the number added at each step) with the first term u₀. Another common mistake is to forget that r can be negative: an arithmetic sequence is not necessarily increasing.