Direction of variation of an arithmetic sequence

Increasing, decreasing or constant?

The direction of variation of an arithmetic sequence depends solely on the sign of the common difference r. We examine the difference between two consecutive terms: u(n+1) - u(n) = r

• If r > 0, the sequence is strictly increasing: each term is greater than the previous one.
• If r < 0, the sequence is strictly decreasing.
• If r = 0, the sequence is constant: all terms are equal.

Example

Let $u_n = 10 - 2n$ (so $u_0 = 10$, $r = -2$). u_0 = 10, u_1 = 8, u_2 = 6, u_3 = 4: the sequence does indeed decrease by 2 at a time, which is consistent with r = -2 < 0.

Why this rule always works

Unlike other sequences, the change in an arithmetic sequence never depends on n: it is the same throughout, from the first to the last term. This is a direct consequence of the definition u(n+1) = u(n) + r: the same quantity is always added.

Sign of r	Behaviour
r > 0	increasing
r = 0	constant
r < 0	decreasing

Pitfalls to avoid

• Do not simply look at the first two terms: here, the rule works precisely because r is constant for all n; this must be justified.
• r = 0 is a valid special case: a constant sequence remains an arithmetic sequence; it is neither increasing nor decreasing, but there is nothing to prevent this.
• Be sure to calculate u(n+1) – u(n) and not u(n) – u(n+1), which would give the opposite sign and lead to the opposite conclusion.