If `X""=""{4^n-3n-1"":""n in N}""a n d""Y""=""{9(n-1)"": n in N}` , where N is the set of natural numbers, then `XuuY` is equal to (1) N (2) Y - X (3) X (4) Y

Options

A. X
B. Y
C. N
D. Y - X

Answer

B

Explanation

Since, `4^(n)-3n-1=(1+3)^(n)-3n-1`
`=(1+.^(n)C_(1).3+.^(n)C_(2).3^(2)+.^(n)C_(3).3^(3)+...+.^(n)C_(n).3^(n))-3n-1`
`=3^(2)(.^(n)C_(2)+.^(n)C_(3).3+...+.^(n)C_(n).3^(n-2))`
`implies 4^(n)-3n-1` is a multiple of 9 for `n ge 2`
For `n = 1, 4^(n) - 3n - 1 = 4 - 3 - 1 = 0`
For `n = 2, 4^(n) - 3n - 1 = 16 - 6 - 1 = 9`
`therefore 4^(n) - 3n - 1` is multiple of 9 for all `n in N`.
It is clear that X contains elements, which are multiples of 9 and Y contains all multiples of 9.
`therefore XsubeY " i.e., " XuuY=Y`