`lim_(ntoinfty) (root3(1)+root3(2)+...+root3(n))/(n^(7//3)((1)/((na+a)^(2))+(1)/((na+2)^(2))+.. .+(1)/((na+n)^(2))))=54` then possible values a is/zer

Options

A. `-9`
B. `8`
C. `7`
D. `-6`

Answer

A, B

Explanation

`underset(ntoinfty)lim((1)/(n)underset(r=1)overset(n)sum((r)/(n))^(1//3))/((1)/(n)((n^(2))/((na+1)^(2))+(n^(2))/((na+2)^(2)). . . (1)/((na+1)^(2))))=`
`(underset(0)overset(1)intx^(1//3)dx)/(int_(0)^(1)(dx)/((a+x)^(2)))=([(3)/(4)x^(4//3)]_(0)^(1))/([-(1)/(a+x)]_(0)^(1))=((3)/(4))/((1)/(a)-(1)/(a+1))=54`
`implies(1)/(a)-(1)/(a+1)=(3)/(4xx54)implies(1)/(a(a+1))=(1)/(72)`
`a=8` or `a=-9`