Question

If $X=8^n-7n-1,n\in NandY=49(n-1),n\in N$, then -

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

Answer 1

The correct option is A $X\subset Y$

Conventional Approach:

For $n=1,8^n-7n-1=8-7-1=0;$
For $n=2,8^n-7n-1=64-14-1=49$
$\therefore 8^n-7n-1$ is a multiple of 49 for all $n\in N.$
$\therefore$ X contains elements which are multiples of 49 and Y clearly contains all multiples of 49.
$\therefore X\subset Y.$

Best Approach:

Put n = 1, X = 0, Y = 0
For n = 2, X = 49, Y = 49
For n = 3, X = 490, Y = 98
$\therefore$ X contains elements which are multiples of 49 and Y clearly contains all multiples of 49.
$\therefore X\subset Y.$