Understanding Induction: How to Prove 1^2+2^2+...+n^2 = n(n+1)(2n+1) / 6

[Sven]

Sorry if this is a really stupid question :( I'm a bit confused on something. I have the whole proof written out and I've checked it everywhere, but I'm confused on WHY for one step.



This is for proving 1^2+2^2+...+n^2 = n(n+1)(2n+1) / 6. So, you add (n+1)^2 and all that. I

http://img199.imageshack.us/img199/6412/mathwtf.jpg
http://img199.imageshack.us/img199/6412/mathwtf.jpg

From the second to the third line...I understand most of the rearrangement...but where did the square go? Why does it go frmo ending with 6(n+1)^2 to ending with 6(n+1)? Where is the square??

Thank you. Also, any tips for induction? I get other problems, but sometimes it is not clear to me at all how I'm supposed to manipulate it to equal what I need it to equal. Any general tips? Sometimes I try to work backwards for inspiration

 

[rock.freak667]

In the third line, after the 6(n+1) put another closing bracket ) and then you will see that they just simply factored out an 'n+1' from the n(2n+1)(n+1) and 6(n+1)².

Well sometimes you do indeed need to work backwards a bit. And in your example you know in induction your final answer for the sum of (N+1) squares would be just replacing the 'n' (in the inductive hypothesis) by 'N+1'

example, say I want to show that 1+2+3+...+n = n(n+1)/2

If I am proving this by induction, I know that my final thing should be


1+2+3+..+N+(N+1) = [N+1]([N+1]+1)/2

 

[Sven]

OHHHHHHHHHH! I see it now! Thank you so much!

And thanks for the tip, I'll definitely keep that in mind =)

Again, thank you so much for sorting out my slow little brain, lol.