Question

Mean of $n$ items is $\overline{x}$. If these $n$ items are successively increased by $2,2^2,2^3,..,2^n$, then the new mean is

• A. $\overline{x}+\frac{2^{n+1}}{n}$
• B. $\overline{x}+\frac{2^{n+1}}{n}-\frac{2}{n}$
• C. $\overline{x}+\frac{2^n}{n}$
• D. $\overline{x}+2^n$

Answer 1

The correct option is B $\overline{x}+\frac{2^{n+1}}{n}-\frac{2}{n}$

We know
$\frac{x_1+x_2+x_3+\cdots +x_n}{n}=\overline{x}$
The new mean
$=\frac{(x_1+2)+(x_2+2^2)+(x_3+2^3)+\cdots +(x_n+2^n)}{n}=\frac{(x_1+x_2+x_3+\cdots +x_n)+(2+2^2+2^3\cdots +2^n)}{n}$
We know,
$2+2^2+2^3\cdots +2^n=\frac{2(2^n-1)}{2-1}$
So, the new mean
$=\overline{x}+\frac{2(2^n-1)}{n}=\overline{x}+\frac{2^{n+1}}{n}-\frac{2}{n}$