# Need help with induction

• Jan 11th 2013, 09:53 AM
borab
Need help with induction
Greetings, I've been trying to solve this question for quite some time now but I just can't figure it out.

http://latex.codecogs.com/gif.latex?...n+1)(2n+1)}{6}

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

Assuming that:
http://latex.codecogs.com/gif.latex?...n+1)(2n+1)}{6} is true for every natural number, I need to prove it for n+1, like this:

http://latex.codecogs.com/gif.latex?...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:

http://latex.codecogs.com/gif.latex?...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?

• Jan 11th 2013, 10:12 AM
Plato
Re: Need help with induction
Quote:

Originally Posted by borab
http://latex.codecogs.com/gif.latex?...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?

$\displaystyle \frac{n(n+1)(2n+1)}{6}+(n+1)^{2}=(n+1)\left[\frac{n(2n+1)+6(n+1)}{6}\right]$
• Jan 11th 2013, 10:29 AM
borab
Re: Need help with induction
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

http://latex.codecogs.com/gif.latex?...n+2)(2n+3)}{6}
• Jan 11th 2013, 11:02 AM
vincisonfire
Re: Need help with induction
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.
• Jan 11th 2013, 11:04 AM
Plato
Re: Need help with induction
Quote:

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

http://latex.codecogs.com/gif.latex?...n+2)(2n+3)}{6}

NO I did not.

That happens to be your job, not mine.

Hint: Multiply $\displaystyle n(2n+1)+6(n+1)$, combine and factor.
• Jan 11th 2013, 12:13 PM
borab
Re: Need help with induction
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?
• Jan 11th 2013, 12:48 PM
MarkFL
Re: Need help with induction
It appears to me that Plato has left you the task of showing:

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