Question

The mean of $n$ terms is $\overline{x}$. If the first term is increased by $1$, second by $2$ and so on, then the new mean is

• A. $\overline{x}+n$
• B. $\overline{x}+\frac{n}{2}$
• C. $\overline{x}+\frac{n+1}{2}$
• D. None of these

Answer 1

The correct option is C

$\overline{x}+\frac{n+1}{2}$

Explanation for the correct option.

Find the new mean.

It is given that, the mean of $n$ terms is $\overline{x}$.

Assume that, the $n$ terms are $x_1,x_2,x_3,.....,x_n$.

Therefore, $\overline{x}=\frac{x_1+x_2+x_3+.....+x_n}{n}...1$.

Since, it is given that the first term is increased by $1$, second by $2$ and so on.

That means, the $n$ terms become $x_1+1,x_2+2,x_3+3,.....,x_n+n$.

So, the new mean $\overline{X}$ is as follows:

$$\begin{gathered} \overline{X}=\frac{x_1+1+x_2+2+x_3+3+.....+x_n+n}{n} \\ \Rightarrow \overline{X}=\frac{\left(x_1+x_2+x_3+.....+x_n\right)+(1+2+3+....+n)}{n} \end{gathered}$$

From equation $1$.

$$\begin{gathered} \Rightarrow \overline{X}=\overline{x}+\frac{(1+2+3+....+n)}{n} \\ \Rightarrow \overline{X}=\overline{x}+\frac{n(n+1)}{2n} \\ \Rightarrow \overline{X}=\overline{x}+\frac{(n+1)}{2} \end{gathered}$$

Therefore, the new mean is $\overline{x}+\frac{n+1}{2}$.

Hence, option $C$ is the correct option.