Given, mean is ,
Let x1, x2, …, xn 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.
Given as the first term is increased by 1 and 2nd term is increased by 2 and so on. Then the terms will be
x1 + 1, x2 + 2, …,xn + n
Let the new mean be x
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  and using equation  we have
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