Question

If $A=\{4^n-3n-1:n\in N\}$ and $B=\left\{9\right(n-1):n\in N\}$, then?

• A. $B\subset A$
• B. $A\cup B=N$
• C. $A\subset B$
• D. None of these

Answer 1

The correct option is A $B\subset A$

$4^n-3n-1,n\in N{\left(3+1\right)}^n-3n-1byu\sin gbinomalension\Rightarrow \left[1+3n{+}^n{C}_2\cdot 3^2{+}^n{C}_3\cdot 3^3+{......}^n{C}_n\cdot 3^n\right]-3n-1\Rightarrow 1+3n+9^n{C}_2+{27}^n{C}_3+...{.3}^n-3n-1\Rightarrow 9\left({}^n{C}_2+3^n{C}_3+{.....}^n{C}_n{3}^{\left(n-2\right)}\right)Whichisdivisibleby\left(q\right),Allelementof\left(B\right)isincludingin\left(A\right)B\subset A$