Question

If $A=\{4^n-3n-1:nϵN\}$ and $B=\left\{9\right(n-1):nϵN\}$, then we have

• A. $A\subset B$
• B. $A\cap B=\varphi$
• C. $B\subseteq A$
• D. None of these

Answer 1

The correct option is A $A\subset B$

For $n=1,4^n-3n-1=4-3-1=0=9(1-1)ϵB$
For $n\ge 4,4^n-3n-1=(1+3{)}^n-3n-1$
$={(}^n{C}_0{+}^n{C}_1\cdot 3{+}^n{C}_2\cdot 3^2+.....{+}^n{C}_n\cdot 3^n)=3n-1$
${=}^n{C}_2\cdot 3^2+.....{+}^n{C}_n\cdot 3^n=9{(}^n{C}_2+....+3^{n-2})$
$=9\{{(}^n{C}_2+.....+3^{n-2}+1)-1\}ϵB$
$\therefore A\subseteq B$.
Using roster method,
$A=\{0,9,54,243,....\}$
and $B=\{0,9,18,27,36,45,54,.....\}$
$\therefore A\subset B.(\because A\ne B)$