Proof by induction question

Hi guys!
I desperately need help on this proofs question. I really am not understanding the process to prove this question.
For n>=2, prove n^2>=2n for all n natural numbers.
I've proven true so far for n=2 and am in the process of proving true for n=k and n=k+1 which has therefore led me to:
k^2+2k+1 = 2k + 2k + 1.
After this, I'm not sure where to go. I've asked everyone around me but everyone seems to be in the dark too. Please help! Sorry if this is not looking aesthetically pleasing.
Thanks.

Reply 1

Shozab_N15

Original post by Oakenari

can u upload the question?

Reply 2

Shozab_N15

Original post by Oakenari

this doesn't even look like a proper question

Reply 3

the bear

we want to show that

( k + 1 )² ≥ 2( k + 1 )

k² + 2k + 1 ≥ 2k + 2

now refer to the assumption step... we can say that k² ≥ 2k,

so the problem changes to showing that

2k + 1 ≥ 2

Reply 4

Oakenari

Original post by Shozab_N15

this doesn't even look like a proper question

This is what needs to be proved.

thumbnail_image_3.png3.8KB

Reply 5

DFranklin

Original post by Oakenari

This is what needs to be proved.

That doesn't say you need to use induction; it's a *lot* easier to prove this directly...

Reply 6

Oakenari

Original post by DFranklin

That doesn't say you need to use induction; it's a *lot* easier to prove this directly...

They asked us to prove this by induction but do tell the more direct method please?

Reply 7

DFranklin

Original post by Oakenari

They asked us to prove this by induction but do tell the more direct method please?

From "

n\geq2

", multiply both sides by n.

Reply 8

Oakenari

Original post by DFranklin

From "

n\geq2

", multiply both sides by n.

If that counts as uni level proof... I'll take it. That makes much more sense!

Reply 9

DFranklin

Original post by Oakenari

If that counts as uni level proof... I'll take it. That makes much more sense!

If you're asked to prove it by induction, then you *have* to do it by induction. At this point they're testing you know how to form an inductive argument, not whether you can prove n^2 >= 2n.

Reply 10

RDKGames

Study Forum Helper

If it's induction, note that

(k+1)^2 = k^2 + 2k + 1 = 2(k+1) + (k^2 - 1)

and just state the (obvious) lower bound for this.

Reply 11

Oakenari

Original post by RDKGames

If it's induction, note that

(k+1)^2 = k^2 + 2k + 1 = 2(k+1) + (k^2 - 1)

and just state the (obvious) lower bound for this.

Right, but my question is how did you get there?

Reply 12

DFranklin

Original post by Oakenari

Right, but my question is how did you get there?

Were you not happy with the solution posted by the bear? (https://www.thestudentroom.co.uk/showpost.php?p=91360154&postcount=4)

(I wouldn't even have posted my solution if I didn't think this had already been resolved).

Reply 13

Oakenari

Original post by DFranklin

I beg your pardon. I clearly wasn't looking through the forum properly. @the bear ... thank you for the solution. That clarifies things a lot more. And to the rest of you, sorry for taking up your time with this question. Just started uni so am a little bit stressed with the online education and so on.

Reply 14

DFranklin

Original post by Oakenari

No problem! I wasn't complaining that you were wasting my time, more that I wouldn't have risked derailing the conversation with an "off-piste" solution if I didn't think the original question was resolved.

Something I'd say about the stress (that I wish *everyone* starting maths at Uni would read and appreciate): no-one expects you to be getting everything right at this point. At this point coursework is for both you and your examiners to get an idea of where you're starting from - it's not going to make any difference to your final degree.