Question

If X = { $8^n-7n-1:nϵN$ } and Y = {$49(n-1):nϵN$}, then prove that $X\subseteq Y$.

Answer 1

$X=\{8^n-7n-1:xϵN\}$

$Y=\left\{4n\right(n-1):nϵN\}$

In order to show that $x\subseteq Y$ we show that every element of X is an element of Y.

So let $xϵX\Rightarrow x=8^m-7^m-1$ for some $mϵN$.

$\Rightarrow x=(1+7{)}^m-7m-1$
= { ${}^m{C}_0{1}^m{+}^m{C}_1{1}^{m-1}7+...{+}^m{C}_{m-1}{1}^1{}_{7}m-1{+}^m{C}_m{7}^m)-7m-1$

[using binomial expansion]$=1+7m{+}^m{C}_2{7}^2{+}^m{C}_3{7}^3+...{.}^m{C}_m{7}^m-7m-1$

${=}^m{C}_2{7}^2{+}^m{C}_3{7}^3+...{.}^m{C}_m{7}^m$

=$49{(}^m{C}_2{+}^{m}m{C}_3+....{+}^m{C}_m{7}^{m-2}),m\ge 2$

$=49t_m,m\ge 2,where{t}_m{=}^m{C}_2{+}^m{C}_{3}7+....{+}^m{C}_m{7}^{m-2}$

Is some positive interger depending on $m\ge 2$

For m = 1

$x=8^1-7\times 1-1$

= 8-8

= 0

Hence, X contains all positive integral multiples of 49.

Also, Y consists of all positive integral multiples of 49, including 0, for n = 1.

Thus, we conclude that $X\subset Y$