(k/k+1)+ (1/(k+1)(k+2)) = (k+1)/(k+1)+1
I just need to make the left side equal to the right side. This is what I have:
k(k+2)/(k+1)(k+2) + 1/(k+1))k+2) =(k+1)/(k+1)+1
k(k+2)+1/(k+1)(k+2) = (k+1)/(k+1)+1
Well the top line in the left hand side of your last equation is...
$\displaystyle k(k+2) + 1 = k^2 + 2k + 1 = (k+1)^2$
So you get
$\displaystyle \frac{(k(k+2)+1)}{(k+1)(k+2)} = \frac{(k+1)^2}{(k+1)(k+2)} = \frac{k+1}{k+2}$.
Which if you want to write it the way you've been doing is...
$\displaystyle \frac{k+1}{(k+1) + 1}$