HSC 2012 MX2 Marathon (archive) (2 Viewers)

[Carrotsticks]

Re: 2012 HSC MX2 Marathon

This is a question from our 4U test this afternoon (I hope I remembered it correctly...):

<a href="http://www.codecogs.com/eqnedit.php?latex=\textup{Prove by induction that }\\ (1-a_{1})(1-a_{2})...(1-a_{n})>1-(a_{1}@plus;a_{2}@plus;...@plus;a_{n})\\ \textup{for positive integers n}\geq 2 \textup{ where}\\ 0<a_{k}<1 \textup{ for }1<k<n" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\textup{Prove by induction that }\\ (1-a_{1})(1-a_{2})...(1-a_{n})>1-(a_{1}+a_{2}+...+a_{n})\\ \textup{for positive integers n}\geq 2 \textup{ where}\\ 0<a_{k}<1 \textup{ for }1<k<n" title="\textup{Prove by induction that }\\ (1-a_{1})(1-a_{2})...(1-a_{n})>1-(a_{1}+a_{2}+...+a_{n})\\ \textup{for positive integers n}\geq 2 \textup{ where}\\ 0<a_{k}<1 \textup{ for }1<k<n" /></a>

haha.

 

[Carrotsticks]

Re: 2012 HSC MX2 Marathon

Well really I suppose more justification is needed etc but meh.

 

[Internet_Police]

Re: 2012 HSC MX2 Marathon

Well really I suppose more justification is needed etc but meh.

not good enough son.

 

[kingkong123]

Re: 2012 HSC MX2 Marathon

haha.

m8.

Well really I suppose more justification is needed etc but meh.

please explain

 

[largarithmic]

Re: 2012 HSC MX2 Marathon

haha.

It's not clear at all why the whole second part thingo after the sum ai is necessarily positive.

 

[Carrotsticks]

Re: 2012 HSC MX2 Marathon

It's not clear at all why the whole second part thingo after the sum ai is necessarily positive.

Haha I know, that's why I said it needs justification, because whether it's a plus or a minus is dependent on what n is.

 

[largarithmic]

Re: 2012 HSC MX2 Marathon

Haha I know, that's why I said it needs justification, because whether it's a plus or a minus is dependent on what n is.

It doesnt just depend on whether its a plus or minus though. Like, you have I think n-2 or n-3 sums after that, and its not clear that given a certain sum it has say, a bigger absolute value, than the one that comes after it for instance. Sorry but I just dont buy the argument at all

 

[Carrotsticks]

Re: 2012 HSC MX2 Marathon

It doesnt just depend on whether its a plus or minus though. Like, you have I think n-2 or n-3 sums after that, and its not clear that given a certain sum it has say, a bigger absolute value, than the one that comes after it for instance. Sorry but I just dont buy the argument at all

We are told that a_k E (0,1), so this raises the issue to prove that:



And it can the inequality follows.

Sorry if I'm being vague. The new *fabulous* Carslaw rooms closes in 5 minutes and I'm already pushing the security guard's patience =D

I'll put a more rigorous proof when I get home and have dinner. Sydney Uni Union sandwiches just don't cut it nowadays.

 

[seanieg89]

Re: 2012 HSC MX2 Marathon

The induction is straightforward.

 

[largarithmic]

Re: 2012 HSC MX2 Marathon

We are told that a_k E (0,1), so this raises the issue to prove that:



And it can the inequality follows.

Sorry if I'm being vague. The new *fabulous* Carslaw rooms closes in 5 minutes and I'm already pushing the security guard's patience =D

I'll put a more rigorous proof when I get home and have dinner. Sydney Uni Union sandwiches just don't cut it nowadays.

I assume the inequality signs are supposed to be the other way; but youve got a problem because there are by no means the same number of summands in sum aiaj and sum aiajak.

 

[math man]

Re: 2012 HSC MX2 Marathon

This is a question from our 4U test this afternoon (I hope I remembered it correctly...):

<a href="http://www.codecogs.com/eqnedit.php?latex=\textup{Prove by induction that }\\ (1-a_{1})(1-a_{2})...(1-a_{n})>1-(a_{1}@plus;a_{2}@plus;...@plus;a_{n})\\ \textup{for positive integers n}\geq 2 \textup{ where}\\ 0<a_{k}<1 \textup{ for }1<k<n" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\textup{Prove by induction that }\\ (1-a_{1})(1-a_{2})...(1-a_{n})>1-(a_{1}+a_{2}+...+a_{n})\\ \textup{for positive integers n}\geq 2 \textup{ where}\\ 0<a_{k}<1 \textup{ for }1<k<n" title="\textup{Prove by induction that }\\ (1-a_{1})(1-a_{2})...(1-a_{n})>1-(a_{1}+a_{2}+...+a_{n})\\ \textup{for positive integers n}\geq 2 \textup{ where}\\ 0<a_{k}<1 \textup{ for }1<k<n" /></a>

this the tech test?

 

[deswa1]

Re: 2012 HSC MX2 Marathon

this the tech test?

Yep

 

[jenslekman]

Re: 2012 HSC MX2 Marathon

mate, how long was teh solution for that?

 

[Carrotsticks]

Re: 2012 HSC MX2 Marathon

 

[math man]

Re: 2012 HSC MX2 Marathon

mate, how long was teh solution for that?

looking at the solution one of my students put forth, not too long

 

[mirakon]

Re: 2012 HSC MX2 Marathon

Overkill lol

 

[Carrotsticks]

Re: 2012 HSC MX2 Marathon

soz carrot 0 marks, had to prove by induction

Haha I know. But the inductive proof as seanieg demonstrated, is very straightforward.

The question is, how DID they get the result?

Suppose I ask you "Give me a bound for the series ", you can't use 'proof by induction' because induction doesn't actually give you the formula or the expression. Rather, it is a means of verifying an identity or a conjecture.

If I wrote a question and received to proofs, one by induction and the other by deriving it from scratch, of course I will consider the latter to have a 'better proof' although both are perfectly valid. Do you know what I'm saying?

I don't like 'proof by induction' because it really should be called 'Verification by induction'.

Similar to Epsilon Delta arguments for the limit. It usually doesn't give you the actual limit, but it lets you VERIFY whether your conjectured limit is actually the limit. You NEED a limit to test otherwise the epsilon/delta proof won't work.

Similarly, you need a formula or expression to prove in order to use induction, otherwise it won't work.

 

[jet]

Re: 2012 HSC MX2 Marathon

Haha I know. But the inductive proof as seanieg demonstrated, is very straightforward.

The question is, how DID they get the result?

Suppose I ask you "Give me a bound for the series ", you can't use 'proof by induction' because induction doesn't actually give you the formula or the expression. Rather, it is a means of verifying an identity or a conjecture.

If I wrote a question and received to proofs, one by induction and the other by deriving it from scratch, of course I will consider the latter to have a 'better proof' although both are perfectly valid. Do you know what I'm saying?

I don't like 'proof by induction' because it really should be called 'Verification by induction'.

Similar to Epsilon Delta arguments for the limit. It usually doesn't give you the actual limit, but it lets you VERIFY whether your conjectured limit is actually the limit. You NEED a limit to test otherwise the epsilon/delta proof won't work.

Similarly, you need a formula or expression to prove in order to use induction, otherwise it won't work.

I think you're greatly underestimating the skill of professional mathematicians Every lecturer I've ever had has mentioned that mathematicians work backwards. They start with a result they think is true and try to find a proof for it.

P.s. You need to have a read over what you have written. Some of it doesn't make sense.

 

[Carrotsticks]

Re: 2012 HSC MX2 Marathon

I think you're greatly underestimating the skill of professional mathematicians Every lecture I've ever had has mentioned that mathematicians work backwards. They start with a result they think is true and try to find a proof for it.

P.s. You need to have a read over what you have written. Some of it doesn't make sense.

Which part for example?

Regarding induction, I understand what you are saying.

Perhaps the HSC has killed my opinion of Induction as it did for many other aspects of education. I haven't done anything extra-curricular, so my only exposure to it was the HSC.

 

[jet]

Re: 2012 HSC MX2 Marathon

Which part for example?

Regarding induction, I understand what you are saying.

Perhaps the HSC has killed my opinion of Induction as it did for many other aspects of education. I haven't done anything extra-curricular, so my only exposure to it was the HSC.

Well the entire bit between "This means that for all..." and "Suppose...". There isn't any other place that you mention a_c so why do you introduce it?