Question

If $X=\{4^n-3n-1|n\in N\}$ and $Y=\{9(n-1)|n\in N\}$, then

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

Answer 1

Answer: (A)

We have, $X=\left\{4^n-3n-1\mid x\in N\right\}$
Now, consider
$4^n-3n-1=(1+3{)}^n-3n-1$
$=3^n+{}^n{C}_{n-1}{3}^{n-1}+\dots +{}^n{C}_2{3}^2+{}^n{C}_{1}3+{}^n{C}_0-3n-1$
(Using Binomial expansion)
$=3^2\left(3^{n-2}+{}^n{C}_{n-1}{3}^{n-3}+\dots +{}^n{C}_2\right)$
$=9\left(3^{n-2}+{}^n{C}_{n-1}{3}^{n-3}+\dots +{}^n{C}_2\right)$
$\Rightarrow$ Set $X$ has natural numbers which are multiples of $9$ (but not all)
Clearly, set $Y$ has all multiples of $9$ .
$\Rightarrow X\subset Y$