If `sum_(i=1)^(n-1) ((""^(n)C_(i-1))/(""^(n)C_(i)+""^(n)C_(i-1)))^(3) = (36)/(13)`, , then n is equal to

Options

A. 10
B. 11
C. 13
D. 12

Answer

d

Explanation

We have ,
`sum_(i-1)^(n-1) ((""^(n)C_(i-1))/(""^(n)C_(i)+""^(n)C_(i-1)))^(3) = (36)/(13)`, ,
`rArr sum_(i-1)^(n-1) ((""^(n)C_(i-1))/(""^(n+1)C_(i)))^(3) = `(36)/(13) ` [ `because ""^(n) C_(r-1) + ""^(n)C_(r) = ""^(n+1)C_(r)]`
`rArr sum_(i-1)^(n-1) ((""^(n)C_(i-1))/((n+1)/(i)""^(n)C_(i-1)))^(3) = (36)/(13) `
`rArr sum_(i-1)^(n-1) ((i)/(n+1))^(3) = (36)/(13)`
`rArr (1)/(n+3)^(3) {(n(n+1))/(2)}^(2) = (36)/(13)`
`(n^(2))/(4 (n +1)^(3))= (36)/(13)`
`rArr (n^(2))/ (n +1)=(144)/(13) rArr (n^(2))/(n+1) = (12^(2))/((12 +1))rArr n=12`.