Question

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

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

Answer 1

The correct option is A $X\subset Y$

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

$=({{}^{n}C}_2{7}^2{+}^n{C}_3{7}^3+...{+}^n{C}_n{7}^n)$

$=49({}^n{C}_2+{7.}^n{C}_3+...+7^{n-2}{.}^n{C}_n)$ for $n\ge 2$

For $n=1,8^n-7n-1=0$

Thus, $8^n-7n-1$ is a multiple of $49$ for $n\ge 2$ and $0$ for $n=1$.

Hence, $X$ consists of all positive integral multiples of $49$ of the form $49K_n$,

where $K_n{=}^n{C}_2+{7.}^n{C}_3+...+7^{n-2}{.}^n{C}_n$ together with zero.

Also, $Y$ consists of all positive integral multiples of $49$ including zero.

Therefore $X\subset Y$

Hence (A) is the correct answer.