If X = {4^n - 3n -1 : n ∈ N} and Y = {9(n - 1) : n ∈ N}

• Binomial Theorem
• Sets

If X = {4ⁿ - 3n -1 : n ∈ N} and Y = {9(n - 1) : n ∈ N}, where N is the set of natural numbers, then X ∪ Y is equal to

1. Y
2. Y - X
3. X
4. N

Answer

Set X contains elements of the form

4ⁿ - 3n - 1 = (1 + 3)ⁿ - 3n - 1

= 3ⁿ + ⁿC_n-1 3^n-1 + ... + ⁿC₂ 3²

= 9(3^n-2 + ⁿC_n-1 3^n-1 + ... + ⁿC₂)

Set X has natural numbers which are multiples of 9 (not all).

Set Y has all multiples of 9.

The correct option is A.