Question

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

• A. X
• B. Y
• C. N
• D. Y - X

Answer 1

The correct option is B Y

$4^n-3n-1=(1+3{)}^n-3n-1={[}^n{C}_0{+}^n{C}_1.3{+}^n{C}_2.3^2+\cdots {+}^n{C}_n{3}^n]-3n-1=9{[}^n{C}_2{+}^n{C}_3.3+\cdots {+}^n{C}_n{.3}^{n-2}]$
$\therefore 4^n-3n-1$ is a multiple of 9 for all $n.$
Therefore, X = {x : x is a multiple of 9}
Also, $Y=9(n-1):n\in N=$ {All multiples of 9}
Clearly $X\subseteq Y\Rightarrow X\cup Y=Y$