Question

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

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

Answer 1

The correct option is B $X\subset Y$

We have
$8^n-7n-1={(1+7)}^n-7n-1$

$={{}^{n}C}_0+{{}^{n}C}_1\times 7+{{}^{n}C}_2{7}^2+....+{{}^{n}C}_n\times 7^n-(7n+1)$

$=7^2({{}^{n}C}_2+{{}^{n}C}_3\times 7+....+{{}^{n}C}_n\times 7^{n-2})$

$\therefore$ $X$ contains some multiples $49$

Clearly $Y$ contains all multiplies of $49$ including $0$.

$\therefore$ $X\subset Y$