Induction Statement (1 Viewer)

[tom.evans.15]

What would be the best induction statement to use (ie the shortest and easiest to remember)?

I've noticed some people say "As it is true for n=1, it is true for n=2, as it is true for n=2, it is true for n=3, therefore it is true for all real n>1" or something similiar, and I've also noticed "Therefore by Mathematical Induction it is true for all n>1"

we were taught the first one, so im not sure about the second, but it seems easier....

 

[study-freak]

What would be the best induction statement to use (ie the shortest and easiest to remember)?

I've noticed some people say "As it is true for n=1, it is true for n=2, as it is true for n=2, it is true for n=3, therefore it is true for all real n>1" or something similiar, and I've also noticed "Therefore by Mathematical Induction it is true for all n>1"

we were taught the first one, so im not sure about the second, but it seems easier....

That's sufficient for the HSC.

 

[Shadowdude]

I believe the shortest is "It is true for n=1, n=k and n=k+1. Therefore, it is true for all n."

But I use: "It is true for n=1, and so true for n=2. Therefore it is true for all n." That's what my teacher taught us... I think the 'so true' makes it sound teenage girly.

 

[Timothy.Siu]

hmm i write

"Since n=1 is true, and n=k+1 is true if n=k is true, then initial statement is true by the principles of mathematical induction"

would i get the marks or do i need to explain that more

 

[jet]

I remember the markers going off in the markers notes about what the student should write. They got pretty pissed because apparently many of the statements people wrote proved that they didn't know anything about induction. I'm pretty sure you're just meant to write the shorter statement study-freak highlighted.

 

[spammy679]

i was taught to write

"the statement is true for n=1, n=2 and so on. Hence it is true for all integers n"

 

[untouchablecuz]

for the way i set my induction proofs i write:

"It follows from parts A and B, by the principle of mathematical induction, that S(n) is true for all [insert condition on n]"

the other conclusion makes induction out to be an iterative proof when in actuality it is not (or so im told)

 

[raniaaa]

since the result is true for n=1, and proven true for n=k and n=k+1, then it is true for all positive integral values of n

 

[untouchablecuz]

hi rania, come on msn

oh wait...

 

[raniaaa]

LOL yeah you're funny =.=
i'm so very very bored

 

[untouchablecuz]

go on webcam

oh wait...

 

[Trebla]

I believe the shortest is "It is true for n=1, n=k and n=k+1. Therefore, it is true for all n."

since the result is true for n=1, and proven true for n=k and n=k+1, then it is true for all positive integral values of n

Technically, these conclusions are incorrect because you didn't actually prove it is true for n = k. You ASSUMED it was true for n = k. Now IF the assumption is true then you proved say n = k + 1 is true, then this means n = k + 1 holds only if n = k holds.

However, you cannot yet say that n = k holds because you've just assumed it. The key is in proving the initial value, say n = 1 without any assumption. Using the "n = k + 1 holds only if n = k holds" statement, THEN you can conclude it is true for n = 2. Since it is true for n = 2, it then holds for n = 3. Since it is true for n = 3, it then holds for n = 4 etc... (bit like a domino effect)

 

[cutemouse]

we were taught the first one, so im not sure about the second, but it seems easier....

You were taught for the 'conclusion' of Step 2 to be "If it is true for n=k, so it is true for n=k+1" and for Step 3 "It is true for n=1 so it is true for n=1+1=2. It is true for n=2 so it is true for n=2+1=3 and so on for all positive integral values of n".

From what I have read in the HSC, it is sufficient to write for Step 3: "Hence it is true for all n1, by induction". Although I think that you must write "If it is true for n=k, so it is true for n=k+1" for the 'conclusion' of Step 2.

Do note that in the HSC they haven't allocated marks for Step 3 (the final conclusion), so it would be a waste of time writing such a long 'mantra' although this year could be different. But on top of that I've read that it's actually wrong to write that long mantra and at university some lecturers will take marks off in an exam if you write that...

I approached the teacher in question about this, but you know that it's hard to change his mind about anything

 

[cutemouse]

Here's the thing I read:



The concluding statements for inductions provided by many candidates show that they incorrectly think that a proof by induction is actually an iterative proof, in which you imagine that the recipe should be repeated as many times as necessary in order to verify the statement for whichever positive integer is of interest.


In fact, the Principle of Mathematical Induction is that every set with the property that, for each integer n in the set, n+1 is also in the set and which also contains 1 contains all positive integers. So, having established that the statement is true for 1 and, if true for some integer, is also true for the next integer, the correct conclusion is to simply state that, by induction, the statement is true for all positive integers. - 2007 HSC Examiners' Report.


I was told by a member of the exam committee that at least 75% of teachers disagree with the exam committee. This is because most maths teachers blindly teach from textbooks written by amateurs without thinking about what they are teaching. Not only is this a very boring way to teach, it also will lead to errors such as the one indicated by the 2007 HSC Examiners' report.


Although this has been corrected in recent years in published solutions to 4 Unit papers, the published solutions to 3 unit papers are still bedevilled by the dreaded mantra, true for n=1, so true for n=2, so true for n=3, etc, therefore true for all positive integers - whereas in 4 unit solutions, we now see by induction, the statement is true for all positive integers.

 

[q3thefish]

Lol I write "Thus it's proven by mathematical induction."

 

[Prashant Sallan]

Lol I write "Thus it's proven by mathematical induction."

+1, we had that maths with the marker video conference thing, he stated what study freak has stated aswell.

 

[cutemouse]

The mantra, ie. "It is true for n=1, so it is true for n=1+1=2. It is true for n=2, so it is true for n=2+1=3 and so on for all positive integral values of n" seems to be the 'old fashioned' way of concluding mathematical induction. I wonder where it actually came from?

 

[EvoRevolution]

Iam not sure if it is for both 3u and 4u but i pretty sure u dont need a concluding statement in 4u induction as no marks are awarded for it.

 

[jet]

Iam not sure if it is for both 3u and 4u but i pretty sure u dont need a concluding statement in 4u induction as no marks are awarded for it.

Refer to Mathematics Extension I 2008 HSC, Question 3 b):

A correct solution implies a complete solution which requires the Induction statement.

 

[cutemouse]

Iam not sure if it is for both 3u and 4u but i pretty sure u dont need a concluding statement in 4u induction as no marks are awarded for it.

You should put it. They may decide to dock you for not putting it any single year.

It's like general solutions. You should put that 'n is any integer', but I'm sure that not many people don't and get away with it. Of course the correct answer would be to put it.