Need help with induction

**borab** · January 11th 2013, 09:53 AM

Greetings, I've been trying to solve this question for quite some time now but I just can't figure it out.

$1^{2}+2^{2}+3^{2}+...+n^{2}=\frac{n(n+1)(2n+1)}{6}$

I proved that it works for n=1, now what I need to do is:

Assuming that:
$1^{2}+2^{2}+3^{2}+...+n^{2}=\frac{n(n+1)(2n+1)}{6}$ is true for every natural number, I need to prove it for n+1, like this:

$1^{2}+2^{2}+3^{2}+...+n^{2}+(n+1)^{2}=\frac{(n+1)(n+2)(2n+3)}{6}$

So I need to prove that what's on the left is equal to the right. Using the assumption I get this:

$\frac{n(n+1)(2n+1)}{6}+(n+1)^{2}=\frac{(n+1)(n+2)(2n+3)}{6}$

So now I need to take whats on the left and make it look like what's on the right, does anyone know how to do it?

Thanks in advance.

**Plato** · January 11th 2013, 10:12 AM

Originally Posted by borab

$\frac{n(n+1)(2n+1)}{6}+(n+1)^{2}=\frac{(n+1)(n+2)(2n+3)}{6}$

So now I need to take whats on the left and make it look like what's on the right, does anyone know how to do it?

$\frac{n(n+1)(2n+1)}{6}+(n+1)^{2}=(n+1)\left[\frac{n(2n+1)+6(n+1)}{6}\right]$

**borab** · January 11th 2013, 10:29 AM

Hey, thanks for the answer, but I didn't really understand what you did there, can you please explain? Also you still didn't reach

$\frac{(n+1)(n+2)(2n+3)}{6}$

**vincisonfire** · January 11th 2013, 11:02 AM

Simply put, you just need to factorize one expression in order to get the next one. Plato suggests that you start by putting the (n+1) factor in front. This is given. Then you can work with a simple quadratic expression.

**Plato** · January 11th 2013, 11:04 AM

Originally Posted by borab

Hey, thanks for the answer, but I didn't really understand what you did there, can you please explain?
Also you still didn't reach

$\frac{(n+1)(n+2)(2n+3)}{6}$

NO I did not.

That happens to be your job, not mine.

Hint: Multiply $n(2n+1)+6(n+1)$ , combine and factor.

**borab** · January 11th 2013, 12:13 PM

I appreciate the help, but I'm afraid I still don't understand what you're telling me to do, it might be because english isn't my first language. Is there any chance you can show the next step?

**MarkFL** · January 11th 2013, 12:48 PM

It appears to me that Plato has left you the task of showing:

$n(2n+1)+6(n+1)=(n+2)(2n+3)$

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