Question

If $X=\left\{8^n–7n–1:n\in N\right\}$ and $Y=\left\{49(n–1):n\in N\right\}$, then

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

Answer 1

The correct option is C

$X\subset Y$

Explanation for the correct option.

Step 1: Find the relation for $x$:

Given that,

$X=\left\{8^n–7n–1:n\in N\right\}$

For, $n=1$,

$\begin{array}{rcl}X& =& 8–7–1\\ & =& 0\end{array}$

Now, for $n=2$,

$\begin{array}{rcl}X& =& 8^2-7\left(2\right)-1\\ & =& 64-14-1\\ & =& 49\end{array}$

Similarly, $n=3,4,5,....$

$X=\left\{0,49,490,...\right\}$

$X$ is a set of numbers divisible by $49$. $...\left(1\right)$

Step 2: Find the relation of $Y$:

It is given that,

$Y=\left\{49(n–1):n\in N\right\}$

For, $n=1$

$\begin{array}{rcl}Y& =& 49(n–1)\\ & =& 0\end{array}$

Now, For $n=2$

$\begin{array}{rcl}Y& =& 49\left(2-1\right)\\ & =& 49\end{array}$

Similarly, for $n=3,4,5,....$

$Y=\left\{0,49,98,...\right\}$

$Y$ is a set of multiples of $49$. $...\left(2\right)$

From the statement $\left(1\right)$ and $\left(2\right)$ we find the relation,

$\Rightarrow X\subset Y$

Hence, option C is correct.