Question

Proof by induction. Summation from 1 to n of (3i^2-i) = n^2(n+1). Thanks for any help.

Answer 1

Okay, can you show the base case is true?

Answer 2

Yes. p(1); 2 = 2

Answer 3

What happens when you plug \(n+1\) into the summation?

Answer 4

I got \[(n+1)^{2}(n+2)\]

Answer 5

No, put it into the summation.

Answer 6

You should get: \[
\sum_{i=1}^{n+1}3i^2-i
\]

Answer 7

RIght. What you just posted is = to what I said. Is that right? Assuming we can prove it

Answer 8

We can't assume that we have proved it.

Answer 9

\[
\sum_{i=1}^{n+1}3i^2-i = 3(n+1)^2-(n+1) +\sum_{i=1}^{n}3i^2-i
\]

Answer 10

\[
=3(n+1)^2-(n+1)+n^2(n+1)
\]

Answer 11

Sorry for the confusion, but that is what I have so far on my paper. I am having trouble with the algebra to prove that what you just posted is equal to \[(n+1)^{2}(n+2)\]

Answer 12

Excuse me, not equal to, but that your equation can be simplified down to the one I just posted.

Answer 13

What does it expand to?

Answer 14

Yours or mine?

Answer 15

What I wrote.

Answer 16

\[3n ^{2}+6n+3-n-1+n ^{3}+n^2\]

Answer 17

Now expand out what it is supposed to become, and you should be able to see how you factor what you currently have into it.

Answer 18

Ok. What I am trying to do is factor in such a way is that my previous post becomes \[(n+1)^2(n+2)\]

Answer 19

Are we on the same page? Just making sure.

Answer 20

Expand out your target and that will let you know how to factor it.

Answer 21

Expanded, it is \[n^2+2n+1+n+2\]

Answer 22

Where to go from here?

Answer 23

http://www.wolframalpha.com/input/?i=3(n%2B1)%5E2%E2%88%92(n%2B1)%2Bn%5E2(n%2B1)
http://www.wolframalpha.com/input/?i=(n%2B1)%5E2(n%2B2)

Both expand out to \(2+5 n+4 n^2+n^3\).
Do you want me to teach you algebra?

Answer 24

I understand that. What I am trying to do right now is simplify\[3(n+1)^2−(n+1)+n^2(n+1) \] down to \[(n+1)^2(n+2)\]
I don't understand how what you just posted will help me get there.

Answer 25

Simplify both expression algebraically

Answer 26

Now when you simplify \((n+1)^2(n+2)\) you will see how to factor it.

Answer 27

Ok. Thank you.