The mean of n items is X¯. 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 items. Then X¯=1nΣxi. Let y1=x1+1,y2=x2+2,y3=x3+3,…,yn=xn+nThen the mean of the new series is 1n∑xi+i=1n∑xi+1n(1+2+3+⋯+n)                =X¯+1nn(n+1)2                =X¯+n+12