To solve $ \frac{1}{a}+\frac{1}{a+b}+\frac{1}{a+b+c}=1$ in natural numbers. I really stuck on the following 7th grade problem (shame on me). 
The problem asks to solve an equation in natural numbers (that is to find all possible such natural  $a,b,c~$ that) 
 $$ \frac{1}{a}+\frac{1}{a+b}+\frac{1}{a+b+c}=1$$ 
I have found solutions: $$a=3, ~b=0, ~c=0;$$ $$a=2, ~b=1, ~c=3$$ and $$a=2, ~b=2, ~c=0$$
but there must be a general trick to find all possible solutions, that I missed.
Can anyone suggest some way to solve the equation? 

The above mentioned equation shown below has parametric solution given below:
$\frac{1}{a}+\frac{1}{a+b}+\frac{1}{a+b+c}=1$
$(a,b,c) =[w(-1-k),w(1+2k),w(k)]$
where,  $w=(k+3)/[2k(k+1)]$

