The sum of the series ` sum_(r=0) ^(n) ""^(2n)C_(r), ` is

Options

A. `n 2^(2n-1)`
B. `2^(2n - 1)`
C. `2^(n-1) + 1`
D. none of these

Answer

a

Explanation

We have,
`rArr sum_(r=1)^(n)r. ""^(2n)C_(r) = sum_(r=1)^(n) r . (2n)/(r) ""^(2n-1)C_(r-1)`
`rArr sum_(r=1)^(n)r. ""^(2n)C_(r) = 2n sum_(r=1)^(n) ""^(2n-1)C_(r-1)`
`rArr sum_(r=1)^(n)r. ""^(2n)C_(r) = 2n {""^(2n-1)C_(0)+""^(2n-1)C_(1)+""^(2n-1)C_(2)+ ...+""^(2n-1)C_(r-1)}`
`rArr sum_(r=1)^(n)r. ""^(2n)C_(r) = n {""^(2n-1)C_(0)+""^(2n-1)C_(1)+""^(2n-1)C_(2)+ ...+""^(2n-1)C_(r-1)+ `
`+ ""^(2n-1)C_(0)+""^(2n-1)C_(n+1)+...+""^(2n-1)C_(2n-2)+""^(2n-1)C_(2n-1)}`
""`[because ""^(n)C_(r) = ""^(n)C_(n-r)]` .
`rArr sum_(r=1)^(n) r. ""^(2n)C_(r) = n (2(2n -1))`