Question

Consider the data on $x$ taking the values $0,2,$ $4,8,\dots ,2^n$ with frequencies ${}^n{C}_0,{}^n{C}_1,{}^n{C}_2,\dots ,{}^n{C}_n$ respectively. If the mean of this data is $\frac{728}{2^n}$, then $n$ is equal to _______.

Answer 1

Answer: 6.00

x	0	2	4	8		$2^n$
f	${}^n{C}_0$	${}^n{C}_1$	${}^n{C}_2$	${}^n{C}_3$		${}^n{C}_n$

Mean $=\frac{\sum x_i{f}_i}{\sum f_i}=\frac{\sum _{r=1}^n{2}^r{}^n{C}_r}{{\sum }_{r=0}^n{}^n{C}_r}$
Mean $=\frac{(1+2{)}^n-{}^n{C}_0}{2^n}=\frac{728}{2^n}$
$\Rightarrow \frac{3^n-1}{2^n}=\frac{728}{2^n}$
$\Rightarrow 3^n=729$
$\Rightarrow n=6$