If mean of n observation is X ̅. If the first observation incereas by 1, the seond by 2, then third by 3 and on, then the new mean is

Correct option is C

Given:

The mean of n observations is X̄. If the first observation increases by 1, the second by 2, the third by 3, and so on, we need to find the new mean.

Explanation:

The sum of the n observations is:

​$$\sum_{i=1}^{n} x_i = n \cdot \bar{X}$$​​

Now, if the first observation increases by 1, the second by 2, and so on, the new sum is:

​$$\text{New sum} = \sum_{i=1}^{n} x_i + (1 + 2 + 3 + \dots + n)$$​​

The sum of the first n natural numbers is:

​$$1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$$​​

So, the new sum is:

​$$\text{New sum} = n \cdot \bar{X} + \frac{n(n+1)}{2}$$​​

The new mean is the new sum divided by n:

​$$\text{New mean} = \frac{n \cdot \bar{X} + \frac{n(n+1)}{2}}{n}$$​​

Simplifying, we get:

​$$\text{New mean} = \bar{X} + \frac{n+1}{2}$$​​

Thus, the correct answer is:

C:

​$$\text{New mean} = \bar{X} + \frac{n+1}{2}$$​​