`lim_(ntoprop) (n)/(3){((3)/(n)+(9)/(n^(2)))^(2)+((3)/(n)+(18)/(n^(2)))^(2)+((3)/(n)+(27)/(n^(2)))^(2)+….+((3)/(n)+(9)/(n))^(2)}` is less than or equal to

Options

A. 62
B. 63
C. 14
D. 21

Answer

A, B, D

Explanation

`underset(nto prop)(lim)sum_(r-1)^(n)(n)/(3)((3)/(n)+(9r)/(n^(2)))^(2)`
`underset(nto prop)(lim)sum_(r=1)^(n)(n)/(3).(9)/(n^(2))(1+(3r)/(n))^(2)`
`underset(nto prop)(lim)sum_(r=1)^(n)(3)/(n)(1+(3r)/(n))^(2)`
`3int_(0)^(1)(1+3x)^(2)dx`
`(cancel(3)(1+3x)^(3))/(cancel(3).3))_(0)^(1)`
`(64)/(4)-(1)/(3)=21`