Question

Consider the data on $x$ taking values $0,2,4,8,...,2^n$ 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

$X_i$	$0$	$2$	$2^2$	$\cdots$	$2^n$
$f_i$	${}^n{C}_0$	${}^n{C}_1$	${}^n{C}_2$	$\cdots$	${}^n{C}_n$

$\overline{X}=\frac{\sum f_i{X}_i}{\sum f_i}$
$=\frac{0\times {}^n{C}_0+2\times {}^n{C}_1+2^2\times {}^n{C}_2+\cdots +2^n\times {}^n{C}_n}{{}^n{C}_0+{}^n{C}_1+\cdots +{}^n{C}_n}$
$=\frac{3^n-1}{2^n}=\frac{728}{2^n}$
$\Rightarrow 3^n=729=3^6$
$\Rightarrow n=6$