Question: Hi can you help me solve this:
I have proved that $p(1)$ is true and am now assuming that $p(k)$ is true. I just don't know how to show $p(k+1)$ for both sides?

Answer: 
Hint: $$\sum\_{r=1}^{k+1}\frac{1}{r(r+1)}=\sum\_{r=1}^{k}\frac{1}{r(r+1)}+\frac{1}{(k+1)((k+1)+1)}.$$ Now use the inductive hypothesis and see if that gets you anywhere.
HINT:

$$\frac1{r(r+1)}=\frac1r-\frac1{r+1}$$

and

$$\frac{n}{n+1}=1-\frac1{n+1}\;.$$