Proving the Sum of Reciprocals by Induction

[thefeedinghan]

Hello there, I'm having trouble proving this by induction

Homework Statement

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

Homework Equations

For the base case n=1

$\frac{1}{1(2)}=\frac{1}{1+1} = \frac{1}{2} = \frac{1}{2}$

$\frac{k}{k+1} + \frac{1}{(k+1)(k+2)}$ <- the second term would be the next integer

The Attempt at a Solution

$\frac{1}{1(2)}+ \frac{1}{2(3)}+\frac{1}{k(k+1)} + \frac{1}{(k+1)(k+2)} = \frac{k+1}{(k+1)+1} = \frac{k}{k+2} + \frac{1}{k+2}$

I don't know where to go from here, any help would be appreciated

 

[tiny-tim]

welcome to pf!

hi welcome to pf!

$\frac{k}{k+1} + \frac{1}{(k+1)(k+2)}$

ok, now put the whole thing over (k+1)(k+2) …

what do you get? ​

 

[thefeedinghan]

oh, thanks very much! was pretty obvious now!