Given, mean is ,

Let x₁, x₂, …, x_n are n observations.

And we know

The mean or average of observations, is the sum of the values of all the observations divided by the total number of observations.

i.e.

…[1]

Given as the first term is increased by 1 and 2^nd term is increased by 2 and so on. Then the terms will be

x₁ + 1, x₂ + 2, …,x_n + n

Let the new mean be x

…[2]

Now, we have series

1, 2, 3, …, n

Clearly the above series is an AP(Arithmetic progression) with

first term, a = 1 and

common difference, d = 1

And no of terms is clearly n.

And last term is also n.

We know, sum of terms of an AP if first and last terms are known is:

Putting the values in above equation we have sum of series i.e.

Using this in equation [2] and using equation [1] we have