How to proof the following without using induction?
This is expected value of a probability question, I found the R.H.S just by looking out the results when m=1,2,3,4,…..
But I cannot find any way to simplify LHS to RHS
For all m, n >= 1
$$
Big{sum_{i=0}^{n-1} {m-1+i choose m-1} (n-i) Big} frac{1}{m+n-1 choose m} = frac{n+m}{m+1}
$$
For example, when m=4,n=3,
$begin{aligned}
L.H.S. &= big{{3 choose 3}(3) + {4 choose 3}(2) + {5 choose 3}(1)big}frac{1}{{6 choose 4}}\
&={(3)(3) + (4)(2) + (10)(1)}/15\
&= 7/5\
R.H.S. &= (4+3)/(4+1) = 7/5
end{aligned}$