Question

If $X=\{8^n-7n-1,n\in N\}$ and $Y=49(n-1)$, $n\in N$, then (given $n>1$)

• A. $X\subset Y$
• B. $Y\subset X$
• C. $X=Y$
• D. $X⊈Y$

Answer 1

The correct option is A $X\subset Y$

Given, $X=8^n-7n-1$
$=(1+7{)}^n-7n-1$
$=1+7n+\frac{n(n-1)}{2}{7}^2+....+7^n-7n-1$
$=\frac{n(n-1)}{2}{7}^2+....+7^n$

$=49[\frac{n(n-1)}{2}+....+7^{n-2}]$

Hence, the set $X$ will be some specific multiples of $49$.

$Y=49(n-1)$
Hence, the set $Y$ will be all multiples of $49$. So, it will contain the elements of $X$ too.

So, $X\subset Y$