Question

If $Y = \{ 49 ( n - 1 ) : n \in N \}$ then

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

Answer 1

Answer: (A)

$X \subseteq Y$

Answer 2

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

=

=

( ${ } ^ { n } C _ { 0 } = { } ^ { n } C _ { n } , { } ^ { n } C _ { 1 } = { } ^ { n } C _ { n - 1 }$ etc.)

= $\therefore$ $8 ^ { n } - 7 n - 1$ is amultiple of 49 for $n \geq 2$.

For $n = 1$ , $8 ^ { n } - 7 n - 1 = 8 - 7 - 1 = 0$; For $n = 2$

$8 ^ { n } - 7 n - 1 = 64 - 14 - 1 = 49$ $\therefore$ $8 ^ { n } - 7 n - 1$ is a multiple of 49 for all

$\therefore$ X contains elements which are multiples of 49 and clearly Y contains all multiplies of 49.

$\therefore$ $X \subseteq Y$.