Payton Rasmussen

2022-10-07

Non-decreasing sequences
I have the following sequence
$\left\{{n}^{4}-6{n}^{2}\right\}$
I have to determine if the sequence is non-decreasing, increasing or decreasing.
In my opinion, the sequence is neither, decreasing, non-decreasing nor increasing because it seems to be increasing for all the terms except for the first and second term. Am I right or I am right?
By the way,is there category for such a sequence? where you have an interval of increasing terms and one of decreasing terms or is such a sequence not even possible?

### Answer & Explanation

bequejatz8d

Step 1
The polynomial factorises as ${n}^{2}\left({n}^{2}-6\right),$, which vanishes somewhere between $n=2$ and $n=3$ if we for the moment think of n as a real variable. It then increases afterwards. Thus, you're right that it cannot be monotonic.
Step 2
Sequences that are not monotonic may generally be classified as oscillating.

Step 1
Let ${a}_{n}={n}^{4}-6{n}^{2}$
Step 2
then ${a}_{n+1}-{a}_{n}=\left(n+1{\right)}^{4}-6\left(n+1{\right)}^{2}-\left({n}^{4}-6{n}^{2}\right)=\dots$

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