If the mean of a set of observations x1,x2,…,xn is X¯ then the mean of the observations xi+2i;i=1,2,…,n is

We have,X¯=r1+x2+…+xnn⇒nX¯=x1+x2+…+xnLet Y be the mean of observations xi+2i;i=1,2,…,n Then Y¯=x1+2⋅1+x2+2⋅2+x3+2⋅3+…+xn+2⋅nn⇒ Y¯=∑i=1n xi+2(1+2+3+…+n)n⇒ Y¯=1n∑i=1n xi+2n(n+1)2n=X¯+(n+1)