Mathematics - Sets and Relations Question with Solution | TestHub

$X=\left\{4^n-3n-1:n\in N\right\}$

$4^n-3n-1={\left(3+1\right)}^n-3n-1$

$={}^n{C}_0{\left(3\right)}^o{1}^n+{}^n{C}_1{\left(3\right)}^1{\left(1\right)}^{n-1}+.....+{}^n{C}_n{\left(3\right)}^n-3n-1$

$=1+n\left(3\right)+{}^n{C}_2{3}^2+.....+3^n-3n-1$

$={}^n{C}_2{3}^2+{}^n{C}_3{3}^3+.....+3^n$

$=3^2\left({}^n{C}_2+{}^n{C}_3{3}^1+{}^n{C}_4{3}^2+.....+3^{n-2}\right)$

$=9k$, where$k$is a natural number.

$Y=\left\{9\left(n-1\right) : n\in N\right\}$

$\therefore X\subseteq Y$

$\therefore X\cup Y=Y$