If `L= lim_(ntooo) (2xx3^(2)xx2^(3)xx3^(4)...xx2^(n-1)xx3^(n))^((1)/((n^(2)+1)))`, and n is even then the value of `L^(4)` is _____________.

Options

A. a. `-(1)/(4)`
B. b. `(1)/(2)`
C. c. 6
D. d. 2

Answer

`(6)`

Explanation

Clearly, n is even. Then,
`underset(ntooo)lim"("2^(1+3+5+...+n//2" terms ").3^(2+4+6+...+n//2" terms")")"^((1)/((n^(2)+1)))`
`=underset(ntooo)lim(2^((n^(2))/(4)).3^((n(n+2))/(4)))^((1)/((n^(2)+1)))`
`=2^(underset(ntooo)lim(1)/(4(1+(1)/(n^(2))))).3^(underset(ntooo)lim((1+(2)/(n)))/(4(1+(1)/(n^(2)))))`
`=2^((1)/(4))3^((1)/(4))=(6)^((1)/(4))`