Question

Consider the data on x taking the values $0,2,4,8,\dots ..,2n$ with frequencies ${}^{n}C_0,{}^{n}C_1,{}^{n}C_2,{}^{n}C_3........,{}^{n}C_n$ respectively. If the mean of this data is $728/2^n$, then $n$ is equal to

Answer 1

Find the value of $n$ :

$X_i$	$0$	$2$	$2^2$	……	$2^n$
$f_i$	${}^{n}C_0$	${}^{n}C_1$	${}^{n}C_2$	……	${}^{n}C_n$

Since the mean of this data is $728/2^n$.

$\begin{array}{rcl}\stackrel{-}{x}& =& \frac{{\displaystyle \sum _{}^{}{f}_i{X}_i}}{{\displaystyle \sum _{}^{}{f}_i}}\\ & =& \frac{0\times {}^{n}C_0+2\times {}^{n}C_1+2^2\times {}^{n}C_2...........2^n\times {}^{n}C_n}{{}^{n}C_0+{}^{n}C_1+............+{}^{n}C_n}\\ & =& \frac{3^n-1}{2^n}\end{array}$

$$\begin{gathered} \Rightarrow \frac{3^n-1}{2^n}=\frac{728}{2^n} \\ \Rightarrow 3^n=729 \\ \Rightarrow 3^n=3^6 \\ \Rightarrow n=6 \end{gathered}$$

Hence the value of $n$is $6$ .