# Proof by induction question

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#1
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.
0
3 months ago
#2
(Original post by Oakenari)
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.
0
3 months ago
#3
(Original post by Oakenari)
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.
this doesn't even look like a proper question
0
3 months ago
#4
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
1
#5
(Original post by Shozab_N15)
this doesn't even look like a proper question
This is what needs to be proved.
0
3 months ago
#6
(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...
0
#7
(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?
0
3 months ago
#8
(Original post by Oakenari)
They asked us to prove this by induction but do tell the more direct method please?
From "", multiply both sides by n.
0
#9
(Original post by DFranklin)
From "", multiply both sides by n.
If that counts as uni level proof... I'll take it. That makes much more sense!
0
3 months ago
#10
(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.
0
3 months ago
#11
If it's induction, note that

and just state the (obvious) lower bound for this.
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#12
(Original post by RDKGames)
If it's induction, note that

and just state the (obvious) lower bound for this.
Right, but my question is how did you get there?
0
3 months ago
#13
(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/sho...54&postcount=4)

(I wouldn't even have posted my solution if I didn't think this had already been resolved).
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#14
(Original post by DFranklin)
Were you not happy with the solution posted by the bear? (https://www.thestudentroom.co.uk/sho...54&postcount=4)

(I wouldn't even have posted my solution if I didn't think this had already been resolved).
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.
0
3 months ago
#15
(Original post by Oakenari)
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.
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.
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