If `X={4^(n)-3n-1:ninN}andy={9(n-1):ninN}`, then `X uu Y` equals
a. `X`
b. `Y`
c. `N`
d. `Y-X`

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`