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If X={4^(n)-3n-1:ninN}andy={9(n-1):ninN}...

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

Text Solution

Verified by Experts

The correct Answer is:
B
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`
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