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March 28, 2024, 10:29:27 pm

Author Topic: Inequality Induction  (Read 5684 times)  Share 

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Lottie99

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Inequality Induction
« on: October 07, 2016, 01:57:03 pm »
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Any tips for how to solve induction inequalities? I seem to always forget how to do it

RuiAce

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Re: Inequality Induction
« Reply #1 on: October 07, 2016, 02:00:52 pm »
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Some brief tips because I'm about to start a tutorial

1. Look at what you're trying to prove, and head in that direction
2. Fudge if necessary
3. Do LHS-RHS<0 if it's easier - optional

Inequalities is the art of throwing out the garbage
« Last Edit: October 07, 2016, 02:02:43 pm by jamonwindeyer »

Lottie99

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Re: Inequality Induction
« Reply #2 on: October 07, 2016, 02:04:12 pm »
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Ah rightio. I just don't really click on the process of the proofs. With classical induction there are the same four steps, but in inequalities I can never manage to incorporate my assumption into the proof of true for n=k+1 properly.

jamonwindeyer

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Re: Inequality Induction
« Reply #3 on: October 07, 2016, 02:06:11 pm »
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Ah rightio. I just don't really click on the process of the proofs. With classical induction there are the same four steps, but in inequalities I can never manage to incorporate my assumption into the proof of true for n=k+1 properly.

Pop us up an example of one you struggle with! It might be easier to explain a frame of reasoning with an example ;D

edmododragon

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Re: Inequality Induction
« Reply #4 on: October 07, 2016, 02:14:57 pm »
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I find that a lot of the inequality inductions require more logical thinking than maths. Sometimes when I do the question I can't make the expression exactly LHS<RHS but I can manage to make it LHS<something and something<RHS therefore LHS<RHS and I end up getting the question that way. Also, I tend to use stuff like (since k is >1, k^2>k+!), stuff that would only work because you're assuming with integers and also because you're only using numbers above a certain integer. Just some things I actively look o ut for and might work for you!

Lottie99

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Re: Inequality Induction
« Reply #5 on: October 07, 2016, 02:15:43 pm »
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This is one from fitzpatrick i think?

edmododragon

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Re: Inequality Induction
« Reply #6 on: October 07, 2016, 02:24:11 pm »
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This is one from fitzpatrick i think?

Interesting, I'm struggling to get this one to work with induction, though I can see an easy way to do it without induction :o

jamonwindeyer

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Re: Inequality Induction
« Reply #7 on: October 07, 2016, 02:24:58 pm »
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Interesting, I'm struggling to get this one to work with induction, though I can see an easy way to do it without induction :o

Yep, this you can do with theory on quadratics, but a solution is coming for induction now! ;D

jamonwindeyer

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Re: Inequality Induction
« Reply #8 on: October 07, 2016, 02:48:28 pm »
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This is one from fitzpatrick i think?

Cool! So I'll assume you are okay with the first test and setting up the assumption, so I'll take it from:



So you are probably okay up to here right? Substitute \(n=k+1\) and simplify. Remember the aim is to prove that this expression is larger than zero, always, using our assumption. The way we do this is purely down to intuition, and can at times seem a little silly, but what we'll do here is this.

We know (or assume) from our assumption that \(k^2-11k+30\). So let's try and make that appear in the expression:



So, using our induction assumption, we've proven the result for \(n=k+1\), provided that \(k\ge5\). This is a little strange for an induction proof, but what it means is that we test \(k=1,2,3,4\) and \(5\) manually, and then induction takes care of the rest.

This is a little atypical for an induction proof, and I don't see any neater way of using induction to do it. You could do something different with the factorisation approach in the latter steps, but you'd end up just doing the typical quadratic solve anyway:



So that seems redundant, hence why I resorted to this weird method. It's more, induction-y ;) This isn't the best example of an induction question imo, I don't blame you for being confused by it, got another? :)

edmododragon

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Re: Inequality Induction
« Reply #9 on: October 07, 2016, 02:51:58 pm »
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Cool! So I'll assume you are okay with the first test and setting up the assumption, so I'll take it from:



So you are probably okay up to here right? Substitute \(n=k+1\) and simplify. Remember the aim is to prove that this expression is larger than zero, always, using our assumption. The way we do this is purely down to intuition, and can at times seem a little silly, but what we'll do here is this.

We know (or assume) from our assumption that \(k^2-11k+30\). So let's try and make that appear in the expression:



So, using our induction assumption, we've proven the result for \(n=k+1\), provided that \(k\ge5\). This is a little strange for an induction proof, but what it means is that we test \(k=1,2,3,4\) and \(5\) manually, and then induction takes care of the rest.

This is a little atypical for an induction proof, and I don't see any neater way of using induction to do it. You could do something different with the factorisation approach in the latter steps, but you'd end up just doing the typical quadratic solve anyway:



So that seems redundant, hence why I resorted to this weird method. It's more, induction-y ;) This isn't the best example of an induction question imo, I don't blame you for being confused by it, got another? :)

That quadratic proof is what I was thinking of. Do you think they would ever ask a question in the HSC where you would need to test more than the initial case (excluding the inductions in Ext2 where each value is dependent on the previous two)?

Lottie99

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Re: Inequality Induction
« Reply #10 on: October 07, 2016, 02:54:10 pm »
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I have an excessive amount of inequalities questions that I dont understand.
Here is another one :)
Thank you so much

RuiAce

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Re: Inequality Induction
« Reply #11 on: October 07, 2016, 02:57:38 pm »
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I have an excessive amount of inequalities questions that I dont understand.
Here is another one :)
Thank you so much
That's gone well beyond MX1.

That's also at the brink of falling out of MX2 territory. I'll work on it later when I have the time.

Lottie99

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Re: Inequality Induction
« Reply #12 on: October 07, 2016, 03:01:55 pm »
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Sorry   ::)

This textbook really doesn't clarify where extension 1 meets extension 2

jamonwindeyer

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Re: Inequality Induction
« Reply #13 on: October 07, 2016, 03:14:51 pm »
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I have an excessive amount of inequalities questions that I dont understand.
Here is another one :)
Thank you so much

Yep that falls outside the range of my abilities ;)

As for the 'testing more than the initial case,' it is unlikely but definitely possible if they put it towards the end of the paper :) 

jakesilove

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Re: Inequality Induction
« Reply #14 on: October 07, 2016, 03:19:12 pm »
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Sorry   ::)

This textbook really doesn't clarify where extension 1 meets extension 2

As a general tip, do the first two steps, do some of the third step, bullshit your way through so it looks like you get an answer, do the last bit, and hope for a lazy marker!
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