The mean of n observations is $\bar x$. If the first observation is increased by 1, second by 2 and so on, then the new mean is :

1. $\bar x$ + n

2. $\bar x$ + $\Big(\dfrac{n}{2}\Big)$

3. $\bar x$ + $\Big(\dfrac{n + 1}{2}\Big)$

4. $\bar x$ + $\Big(\dfrac{n − 1}{2}\Big)$

Let the n observations x₁, x₂, ……., x_n. The mean is $\bar x$.

$\bar{x} = \dfrac{x$1 + x2 + … + x_n}{n} \\[1em] \sum x = n\bar x

Given,

The first observation is increased by 1, second by 2 and so on.

New sum of observations = (x₁ + 1)+ (x₂ + 2)+ …….+ (x_n + n)

= (x₁+ x₂+ …….+ x_n) + (1 + 2 + ….. + n)

= ∑x + (1 + 2 + 3 + … + n)

$= n\bar x + \dfrac{n(n + 1)}{2}$

$\text{ Mean }= \dfrac{\text{ Sum of observations}}{\text{ Number of observations}} \\[1em] = \dfrac{n\bar x + \dfrac{n(n + 1)}{2}}{n} \\[1em] = \bar x + \dfrac{n + 1}{2}.$

Hence, option 3 is the correct option.