How to tell if a sequence is increasing, decreasing, or nonmonotonic

A few terms can give us an idea of where a sequence is headed. To prove its direction, we’ll compare consecutive terms and see whether the same inequality holds at every step.

Compare a term with the next term

A sequence is nondecreasing if $a_{n + 1} \geq a_{n}$ for every index $n$ , and nonincreasing if $a_{n + 1} \leq a_{n}$ for every index. Either condition makes it monotonic. Use “strictly increasing” or “strictly decreasing” when the inequality is strict at every step.

A plateau is allowed in either non-strict direction. To show a sequence is nonmonotonic, find at least one step where it rises and one step where it falls. The two steps can happen at different places.

Example with an increasing sequence

For $a_{n} = n^{3} + 9$ , the first four terms are $10$ , $17$ , $36$ , and $73$ . To prove the increase continues, subtract the current term from the next term.

\begin{array}{l} a_{n + 1} - a_{n} = {(n + 1)}^{3} - n^{3} \\ = 3 n^{2} + 3 n + 1 \end{array}

That difference is positive for every $n \geq 1$ , so the sequence is strictly increasing.

Example with a decreasing sequence

For $a_{n} = \frac{1}{2 n + 3}$ , the denominator grows by $2$ each time, and both denominators are positive. A larger positive denominator gives a smaller reciprocal.

\begin{array}{l} a_{n + 1} = \frac{1}{2 n + 5} < \frac{1}{2 n + 3} = a_{n} \end{array}

So the sequence is strictly decreasing. For positive terms, another way to make the comparison is to divide $a_{n + 1}$ by $a_{n}$ . A ratio below $1$ at every step proves decrease. Here the ratio is $\frac{2 n + 3}{2 n + 5}$ .

Example that decreases only after the first step

Suppose $a_{1} = 0$ and $a_{n} = \frac{1}{n}$ for $n \geq 2$ . The sequence first rises from $0$ to $\frac{1}{2}$ , then falls to $\frac{1}{3}$ and continues to decrease.

These are separate terms. The rise from term 1 to term 2 and the drop from term 2 to term 3 rule out monotonicity.

This sequence is eventually decreasing because every step from the second term onward goes down. It’s nonmonotonic on its full index set because the first step goes up.

For the next step, see how bounds and monotonicity work together.

Learn mathKrista KingJune 1, 2021math, learn online, online course, online math, calculus ii, calculus 2, calc ii, calc 2, sequences and series, sequences, series, infinite series, increasing sequences, decreasing sequences, monotonic sequences, non-monotonic sequences, sequences that are monotonic

How to tell if a sequence is increasing, decreasing, or nonmonotonic

Compare a term with the next term

Example with an increasing sequence

Example with a decreasing sequence

Example that decreases only after the first step

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