If the first item is increased by 1, second by 2 and so on, then the new mean is

Let x1,x2,…,xn be n values of variable X. ThenX¯=1nΣxiLet y1=x1+1,y2=x2+2,y3=x3+3,…,yn=xn+n Then the mean of the new series is given byX′¯=1n∑yi⇒ X′¯=1n∑i xi+i⇒ X′¯=1n∑i xi+1n(1+2+3+…+n)⇒ X′¯=X¯+1n⋅n(n+1)2=X¯+n+12