If n is an even natural number , find the value of
` sum_(r=0)^(n) ((-1)^(r))/(""^(n)C_(r))`

Options

A. 0
B. `(1)/(n)`
C. ` (( -1)^(n//2))/(""^(n)C_(n//2))`
D. none of these

Answer

d

Explanation

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