Question

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

• A. N
• B. Y-X
• C. X
• D. Y

Answer 1

Answer: (D)

$X=\left\{(1+3)^{n}-3 n-1, n \in N\right\}$
$=3^{2}({ }^{n} C_{2}+{ }^{n} C_{3} \cdot 3+\ldots+3^{n-2}), n \in N\} $
$=\{\text { Divisible by } 9\}$
$Y=\{9(n-1), n \in N\} $
$=(\text { All multiples of } 9\} $
So, $X \subseteq Y $
i.e., $X \cup Y=Y $