Question

I was given an assignment to prove that n < or = 3 (n/3) for all nonnegative integers n.

I suspect that this may be a typo, as isn't 1 > 3 ^ (1/3)?

Answer 1

no? It isn't.

Answer 2

\[\Large 3^{\frac{1}{3}} = 1.44224957\]

Answer 3

It should read \[n \ge3^{n/3}\]

Answer 4

Yeah, I totally mixed my signs around.

Answer 5

Yeah, testing the first few values for n gives some interesting results.
\[\Large 3^{\frac{2}{3}} \approx 2.08008382\]
\[\Large 3^{\frac{3}{3}}=3\]
\[\Large 3^{\frac{4}{3}} \approx 4.32674871\]

Answer 6

My original post was backwards of what it should have been. They want me to prove that n≥3^(n/3), which I feel isn't possible. As 1 < 3(n/3)

Answer 7

And then the function grows really rapdily from there.

Answer 8

Oooh, well then yeah. That's an issue.

Answer 9

Haha. Well, yeah. How do I give a medal here?

Answer 10

Click best answer if ya want.

Answer 11

But I didn't do a lot haha. I'm trying to think of how to prove the correct statement...

Answer 12

It states specifically for every Nonnegative Integer, n.

Answer 13

Both 0 and 1 prove it false, so...

Answer 14

Oh wait... I'm so stupid.

Answer 15

Oh, but I mean how I would prove the actual statement.
\[n <= 3^{\frac{n}{3}}\]

Answer 16

Probably pmi.

Answer 17

Just use Well Ordering.

Answer 18

Well Ordering? Never heard of it.

Answer 19

I'll prove it right here, then!

Answer 20

Let C be the set of nonnegative integers for which our theorem doesn't hold true.

Answer 21

For contradiction, let's say that C isn't empty and that it has an element m.

Answer 22

I will make the claim that as C is a set of non-negative integers, there has to be a least element somewhere, and m is that least element.

Answer 23

It's pretty simple to see though that the left side is increasing by 1 each time, while the right side is being multiplied by \[3^{\frac{n-1}{3}}\] which is an increase of greater than 1.

Answer 24

We know that m is > 0 as 0 <= 3 ^0.

Answer 25

right. That's the crux of my proof. :-D

Answer 26

Okay cool. Looks good so far.

Answer 27

We know that m-1 <= 3 ^((m-1)/3), as m is the least element of C.

Answer 28

We know that 3^(1/3) > 1, so we can infer that 3^((m-(m1))/3) > 1