Question

If $X = \{ 4^n - 3n - 1 | n \in N \}$ and $Y = \{ 9(n - 1) | n \in N \}$, then

• A. $ X \subset Y $
• B. $ Y \subset X $
• C. $ X = Y $
• D. None of these

Answer 1

Answer: (A)

We have, $X=\left\{4^{n}-3 n-1 \mid x \in N\right\}$
Now, consider
$4^{n}-3 n-1=(1+3)^{n}-3 n-1 $
$=3^{n}+{ }^{n} C_{n-1} 3^{n-1}+\ldots+{ }^{n} C_{2} 3^{2}+{ }^{n} C_{1} 3+{ }^{n} C_{0}-3 n-1$
(Using Binomial expansion)
$=3^{2}\left(3^{n-2}+{ }^{n} C_{n-1} 3^{n-3}+\ldots+{ }^{n} C_{2}\right) $
$=9\left(3^{n-2}+{ }^{n} C_{n-1} 3^{n-3}+\ldots+{ }^{n} C_{2}\right)$
$\Rightarrow$ Set $X$ has natural numbers which are multiples of $9$ (but not all)
Clearly, set $Y$ has all multiples of $9$ .
$\Rightarrow X \subset Y$