The vaule of ` sum_(r=0)^(n-1) (""^(C_(r))/(""^(n)C_(r) + ""^(n)C_(r +1))` is equal to

Options

A. `(n)/(2) `
B. `(n+1)/(2)`
C. `(n(n+1))/(2)`
D. `(n(n-1))/(2(n+1))`

Answer

a

Explanation

We have, `sum_(r=0)^(n-1) (""^(n)c_(r))/(""^(n)c_(r)+ ""^(n)c_(r+1))=sum_(r=1)^(n-1) (""^(n)c_(r))/(""^(n+ 1)c_(r+1))`
`= sum_(r=0)^(n-1) (""^(n)c_(r))/((n+1)/(r+1) ""^(n)c_(r)) = sum_(r=0)^(n-1) (r+1)/(n+1)`
` = (1)/(n+1)[1+ 2 + ...+ n] = (n(n+1))/(2(n+1)) = (n)/(2)`