Payton Rasmussen

2022-10-07

Non-decreasing sequences

I have the following sequence

$\{{n}^{4}-6{n}^{2}\}$

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?

I have the following sequence

$\{{n}^{4}-6{n}^{2}\}$

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?

bequejatz8d

Beginner2022-10-08Added 6 answers

Step 1

The polynomial factorises as ${n}^{2}({n}^{2}-6),$, 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.

The polynomial factorises as ${n}^{2}({n}^{2}-6),$, 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.

hifadhinitz

Beginner2022-10-09Added 2 answers

Step 1

Let ${a}_{n}={n}^{4}-6{n}^{2}$

Step 2

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

Let ${a}_{n}={n}^{4}-6{n}^{2}$

Step 2

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

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