Induction Inequality

[kemi]

Greetings! Just registered, as I was intrigued by the friendly vibes coming from these forums

Rather simple question:

Prove the cube of the sum of consecutive integers is divisible by 3, where n ∈ N

At the 'assumption' step the proof has already become blatant - I am able to factor out 3. For some reason I feel this is wrong, since the point of the assumption is to use it for the proof. Even upon reaching the 'proof' i.e. prove true for n = k+1, it seems the assumption is unnecessary? Maybe I have made an error in setting up the q. Help would be much appreciated

Thank you!

[jamonwindeyer]

Greetings! Just registered, as I was intrigued by the friendly vibes coming from these forums

Rather simple question:

Prove the cube of the sum of consecutive integers is divisible by 3, where n ∈ N

At the 'assumption' step the proof has already become blatant - I am able to factor out 3. For some reason I feel this is wrong, since the point of the assumption is to use it for the proof. Even upon reaching the 'proof' i.e. prove true for n = k+1, it seems the assumption is unnecessary? Maybe I have made an error in setting up the q. Help would be much appreciated

Thank you!

Hey kemi! Welcome to the forums! Sending some good vibes your way

What you've found is correct - This is an example of a mathematical fact that is just self apparently true. This is reflected in the fact that your assumption is true. So although you can use induction to do it, it's blatantly unnecessary (as you've seen). You should still be able to do the steps, even though we know you don't really need to.

Assume \(k+k+1+k+2=3M\implies3k+3=3M\)
So \(k+1+k+2+k+3=3M+3=3(M+1)\)

That's the rough working. You can still do it - Even though you don't need to

[RuiAce]

Hey kemi! Welcome to the forums! Sending some good vibes your way

What you've found is correct - This is an example of a mathematical fact that is just self apparently true. This is reflected in the fact that your assumption is true. So although you can use induction to do it, it's blatantly unnecessary (as you've seen). You should still be able to do the steps, even though we know you don't really need to.

Assume \(k+k+1+k+2=3M\implies3k+3=3M\)
So \(k+1+k+2+k+3=3M+3=3(M+1)\)

That's the rough working. You can still do it - Even though you don't need to

CUBES, Jamon!!!

[RuiAce]

(0 is the smallest natural number.)





________________

Also, this wasn't really an inequality induction

[de]

I realise that this is supposed to be done with induction. However, I'll just mention a smoother method. You can just consider the equation modulo 3. Then you can sub 0,1,2 into the equation and since and and we are done.

*modulo means considering all numbers as their remainder when divided by 3.

[jamonwindeyer]

CUBES, Jamon!!!

Lol whoops

[RuiAce]

I realise that this is supposed to be done with induction. However, I'll just mention a smoother method. You can just consider the equation modulo 3. Then you can sub 0,1,2 into the equation and since and and we are done.

*modulo means considering all numbers as their remainder when divided by 3.

Modulo arithmetic is not taught in the HSC maths courses.

Else I would've used it.


Explanation to the curious ones though:
If you think about it. If you go back to Year 3 and you did long division with integers, if you divided something by 3 your remainder would be either 0, 1 or 2. Hence, every number must be said to be 0, 1, or 2 "modulo 3". The idea is to think only in terms of the remainder as it significantly reduces your cases.

It's one of the instances where proof by exhaustion is swifter than proof by induction.

[de]

The idea is to think only in terms of the remainder as it significantly reduces your cases.

It's one of the instances where proof by exhaustion is swifter than proof by induction.

Where "significantly" takes you from infinite to 3 cases (or even one if you think about it correctly)!

[RuiAce]

It's good for looking at and learning something extra.

But if you used it in an exam, you'd lose marks. The HSC has become stricter over you sticking to only content taught in the course. Just remember that.

[de]

It's good for looking at and learning something extra.

But if you used it in an exam, you'd lose marks. The HSC has become stricter over you sticking to only content taught in the course. Just remember that.

DW I'm from vic and have already finished! If we avoid discussing stuff external to the course on here we do foster an unnecessarily inward looking perspective. Remember I did provide the disclaimer of "this is supposed to be done with induction".

Also, in vic at least I was under the impression you can put down whatever is correct... providing you explain it properly (unless of course it specifies a method).

[jamonwindeyer]

DW I'm from vic and have already finished! If we avoid discussing stuff external to the course on here we do foster an unnecessarily inward looking perspective. Remember I did provide the disclaimer of "this is supposed to be done with induction".

Also, in vic at least I was under the impression you can put down whatever is correct... providing you explain it properly (unless of course it specifies a method).

In the HSC the same goes, but you won't get marks for working if your answer is incorrect. So if you happen to make a small algebraic error you've lost all the marks. It's a bit of a shame really; I like seeing new methods that work and they lose out because of a sign error!

[RuiAce]

In the HSC the same goes, but you won't get marks for working if your answer is incorrect. So if you happen to make a small algebraic error you've lost all the marks. It's a bit of a shame really; I like seeing new methods that work and they lose out because of a sign error!

Trust me. If the HSC is against L'Hopital's rule, they'll be against anything

(Unless the new 'method' is just a new way of applying techniques taught in the course)

[jamonwindeyer]

Trust me. If the HSC is against L'Hopital's rule, they'll be against anything

(Unless the new 'method' is just a new way of applying techniques taught in the course)

Usually in the cases where an abstract method is used - Provided it is reasonable and correct, it gets the marks. Not sure why L'Hopital's causes such trouble

[RuiAce]

Usually in the cases where an abstract method is used - Provided it is reasonable and correct, it gets the marks. Not sure why L'Hopital's causes such trouble

It used to be fine; L'Hopital's was mentioned and not frowned upon explicitly in the notes from the marking centre.

It's a seemingly recent trend; only came about like 3 years or so ago. Maybe a bit more though. But basically the questions that the exam-writers made were not made for a "tactic" or something outside the scope of the syllabus to be used. Exam writers have to think about what an MX2 student is supposed to do and write questions accordingly; using something like L'H sorta defeats the purpose. Mainly cause they want to assess your understanding of the course you were given, not necessarily your own skills

If they can't pay L'Hopital's rule, I can't see them paying modulo arithmetic either really.


(Not that I approve of it either tbh. But things are just the way they are. Stupid HSC)

Spoiler

It's also a part of why I looked forward to maths at uni back then. Get out of these constraints

[jamonwindeyer]

It used to be fine; L'Hopital's was mentioned and not frowned upon explicitly in the notes from the marking centre.

It's a seemingly recent trend; only came about like 3 years or so ago. Maybe a bit more though. But basically the questions that the exam-writers made were not made for a "tactic" or something outside the scope of the syllabus to be used. Exam writers have to think about what an MX2 student is supposed to do and write questions accordingly; using something like L'H sorta defeats the purpose. Mainly cause they want to assess your understanding of the course you were given, not necessarily your own skills

If they can't pay L'Hopital's rule, I can't see them paying modulo arithmetic either really.

(Not that I approve of it either tbh. But things are just the way they are. Stupid HSC)

Eh, not that it matters because they'll request induction anyway