# Thread: Monotone sequence - how can this be a monotone sequence if it bounces between numbers

1. ## Monotone sequence - how can this be a monotone sequence if it bounces between numbers

My understanding of showing a sequence is monotone is to show that it either increases or decreases by working out

$\displaystyle (a_n+1) - (a_n)$

and if it's > 0 its increasing, if less than, decreasing, right?

But now i have this problem that I must say if the sequence is monotone:

$\displaystyle (a_n) = (-1)^n + 2n$

which to me works out that it is bounded by 0 and 4, ie that it bounces between 0 and 4 ( if n is odd then the value is 0 if even then its 4)

Surely this goes against monotone sequence??? But my answer says that it IS monotone. How can it be monotone when it is neither increasing nor decreasing??

2. A monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing.
You have a monotonically increasing sequence since $\displaystyle a_{n+1}-a_n \geq 0$, but it's not strictly monitonically increasing.

3. Thanks, I was just replying as I picked up my error (I was confusing the $\displaystyle a_n + 1 - a_n$ with the original sequence. Ok I see it is increasing.

But what do you mean by "it is not strictly monotonically increasing?"

Thank you

4. In order for it to be strictly monitonically increasing, you need $\displaystyle a_{n+1} - a_n > 0$