Can this be proven by induction? Because I really can't.

I come down to

k^3 - 3k^2 + 5k = 2k^3 + 3k^ + 3k + 1 which is not equal. :( thanks.

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- July 11th 2013, 04:29 PMjpab29Proving by induction help
Can this be proven by induction? Because I really can't.

I come down to

k^3 - 3k^2 + 5k = 2k^3 + 3k^ + 3k + 1 which is not equal. :( thanks.

Attachment 28783 - July 11th 2013, 05:12 PMHallsofIvyRe: Proving by induction help
You are trying to prove that ?

Your fundamental problem is that this**isn't true**!

When n= 2, while . - July 11th 2013, 05:14 PMjohngRe: Proving by induction help
Hi,

"Plug in" n = 1. This then reads 1 = 1, good so far. Next try n = 2: 1 + 3 = 1^{3}+ 2^{3}, which is of course false. So your formula is not true for all n and so certainly can't be proved by induction.

The above is the correct formula and can be proved by induction. I suggest you try it.

Edit:

It suddenly occurred to me that you probably copied the problem incorrectly. What is true:

This formula is amenable to induction. Try it. - July 11th 2013, 05:31 PMSorobanRe: Proving by induction help
Hello, jpab29!

There must be typo . . .

Quote:

Can this be proven by induction? . No!

. . . This is not true!

- July 11th 2013, 07:22 PMjpab29Re: Proving by induction help
oh that's why. haha Actually it is derived from a conjecture that I am working on.

It is from this link.

http://mathhelpforum.com/discrete-ma...induction.html - July 11th 2013, 07:24 PMjpab29Re: Proving by induction help
Thank you so much for taking the time! I actually have no idea how to work it out. I am taking math class online and it's terrible, no help at all, except from forums like these. I'll try the edit that you've made.

- July 11th 2013, 07:28 PMjpab29Re: Proving by induction help
This is what I'm trying to work on and also trying to understand it through internet's help. :(

Attachment 28790

This is the book example i'm trying to follow.

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Attachment 28792 - July 11th 2013, 07:31 PMMarkFLRe: Proving by induction help
As I posted at MMF, the conjecture you want is:

edit: If you want, I can walk you through how I came up with it. :D - July 11th 2013, 07:43 PMjpab29Re: Proving by induction help
oh okay, so the 2n-1 with the summation sign is not a constant formula? it changes? So sorry about being clueless. The book only gives one example per kind of question. :(

- July 11th 2013, 07:47 PMjpab29Re: Proving by induction help
Thanks so much, yes I'll need a walkthrough :) :) :)

- July 11th 2013, 08:16 PMMarkFLRe: Proving by induction help
This is what I noticed:

For each line 1) - 4) (and letting the line number be ), I noticed that the first number in the sum is:

Now, we then are adding the next consecutive integers. As you observed, the sum is shown to be equal to . So, we may state this as:

Does this make sense? - July 11th 2013, 08:42 PMjpab29Re: Proving by induction help
I still only understood (n-1)^3 + n^3...

But I don't get the left side still... how it evolved. But I tried plugging the N's.

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:( - July 11th 2013, 08:57 PMMarkFLRe: Proving by induction help
Using the conjecture I stated originally (I had a typo the second time...):

We have for:

- July 11th 2013, 09:35 PMjpab29Re: Proving by induction help
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Thank you so much! - July 11th 2013, 10:07 PMMarkFLRe: Proving by induction help
If I was going to prove this by induction, this is what I would do. We have already shown the base case is true, so the induction hypothesis is the so-called conjecture:

Let's simplify a bit:

As the inductive step, I would add :

We have derived from thereby completing the proof by induction.