Induction Method

[k1ll3rdr4g0n]

Ok so my math teacher told me I had to prove this:
For n >= 1, n^3 - n is divisible by 3.

She said that we had to use any method we learned in class. Me and my friend chose the inductive method.

P(1) makes sense.

I just can't figure out P(n + 1) :/.

[NotLKH]

So:
P(1) is true
Assume P(N) is true, that is (N³-N) is divisible by 3 for some N

Then, is P(N+1) divisible by 3?

Well (N+1)³ - (N+1) becomes

N³+3*N² + 3*N + 1 - N - 1

Which, rearranged becomes:

(N³-N) + 3*N² + 3*N {the 1's cancel}

Now, we know (N³-N) is divisible by 3, per our Assumption.
And it is apparent that 3*N² + 3*N is divisible by 3.

So it would appear that, if P(N) is divisible by 3, then P(N+1) is also divisible by 3.


Therefore, since we know P(1) | 3, then also P(2) since 2 = 1+1, and so is P(3) {3 = 2+1}, and so on for all numbers >= 1,
all because we showed that, if P(N) | 3,
then so is P(N+1) | 3

[eranga262154]

Which, rearranged becomes:

(N³-N) + 3*N² + 3*N {the 1's cancel}

Now, we know (N³-N) is divisible by 3, per our Assumption.
And it is apparent that 3*N² + 3*N is divisible by 3.

So it would appear that, if P(N) is divisible by 3, then P(N+1) is also divisible by 3.

Fine

I think the above part will be missd-out by k1ll3rdr4g0n

[k1ll3rdr4g0n]

So:
P(1) is true
Assume P(N) is true, that is (N³-N) is divisible by 3 for some N

Then, is P(N+1) divisible by 3?

Well (N+1)³ - (N+1) becomes

N³+3*N² + 3*N + 1 - N - 1

Which, rearranged becomes:

(N³-N) + 3*N² + 3*N {the 1's cancel}

Now, we know (N³-N) is divisible by 3, per our Assumption.
And it is apparent that 3*N² + 3*N is divisible by 3.

So it would appear that, if P(N) is divisible by 3, then P(N+1) is also divisible by 3.


Therefore, since we know P(1) | 3, then also P(2) since 2 = 1+1, and so is P(3) {3 = 2+1}, and so on for all numbers >= 1,
all because we showed that, if P(N) | 3,
then so is P(N+1) | 3

Thank you so much, that didn't make sense to me until now. I made a little mistake in my algebra and it made me so confused. And my teacher doesn't even know what shes talking about .

[NotLKH]

...And my teacher doesn't even know what shes talking about .

Why do you say that?
What did she say?

[k1ll3rdr4g0n]

Well..
1. Only 2 people out of the 8 people are getting higher than a C
2. She doesn't double check her stuff, she marked something on my test that was right, but she wrote the answer like less than 3 inches away from my answer. Exactly the same. In fact mine was probably more right than hers because she didn't put the base for the first number (it was base conversion) (it was base 10), I did.
3. She expects us to allready know most of the stuff going into her class so it has just been a uphill battle thus far with her.
4. I got an A in all of my math classes in HS, but this class I am barley passing. Tells yeah something doesn't it?
5. She makes you feel stupid if you don't know the answer or say the wrong answer.

[NotLKH]

Hmmm.

Is this class pretty much a general maths class in College/Uni?

From your statement, it seems that you've graduated from High School.

Can you describe what areas of math that was covered in HS?

I ask, because I also was a straight A student in Maths in HS, and we covered everything from trig, proofs, geometry, and In my last year I went into ITCM, {Introduction to College Mathematics}, which was a combination of PreCalc {Learning the basic theory of Calc in terms of Limits of Delta X as X approaches Zero}, and then true Calculus, learning the actual integration and differentiation methods. When I hit college, They almost put me in Calc 3, but there were a couple of things I needed to pick up in Calc2 first.

{Don't ask me about Calculus now, its been 25 years since I last did it. Ohh, I can differentiate and Integrate the most basic polynomials, but I've lost all the identities of Triginometric Calculus.}

[Glaysher]

Well..

4. I got an A in all of my math classes in HS, but this class I am barley passing. Tells yeah something doesn't it?

Not really. If you are from my country with our grading system it would sound about right (and good in a lot of cases) as it would sound like you got an A at GCSE and are struggling with Further Maths A level. This would be expected as you wouldn't have the top grade at GCSE abd I have met many students with the top grade at GCSE that would not be able to cope with the advanced material in A level Further Maths.

But since I don't know where you are from and what you are studying it doesn't tell me anything.

[k1ll3rdr4g0n]

Hmmm.

Is this class pretty much a general maths class in College/Uni?

From your statement, it seems that you've graduated from High School.

Can you describe what areas of math that was covered in HS?

I ask, because I also was a straight A student in Maths in HS, and we covered everything from trig, proofs, geometry, and In my last year I went into ITCM, {Introduction to College Mathematics}, which was a combination of PreCalc {Learning the basic theory of Calc in terms of Limits of Delta X as X approaches Zero}, and then true Calculus, learning the actual integration and differentiation methods. When I hit college, They almost put me in Calc 3, but there were a couple of things I needed to pick up in Calc2 first.

{Don't ask me about Calculus now, its been 25 years since I last did it. Ohh, I can differentiate and Integrate the most basic polynomials, but I've lost all the identities of Triginometric Calculus.}

It's a college course, it is called discrete math. I have to take it unfortunately for my major (computer science). I went from Algebra 1 -> Geometry -> Advanced Algebra - > College algebra trig. I had a chose when I started college between Discrete and calc 1...should've taken calc 1.

I am a freshman in college btw.

[Glaysher]

I never met a teacher at university, only lecturers. 8 people on a university course! Wow more like 500 on a standard maths course at my uni. Calculus 1 would probably be easier

[k1ll3rdr4g0n]

Yeah this is a private university, so class sizes are generally smaller for classes required for certain majors.

I am generally one on one with most of my teachers. All know me by name pretty much. Especially my comp sci teacher.

[eranga262154]

Yeah this is a private university, so class sizes are generally smaller for classes required for certain majors.

I am generally one on one with most of my teachers. All know me by name pretty much. Especially my comp sci teacher.

Fine.

In my University most of the lectures are not like your teachers. If we like to do whatever they ask from us, it is fine. Otherwise they don't care about us.

[Mattywoo2]

What is a private university?

[eranga262154]

What is a private university?

Simply, You have to pay for what you gain from there.