Statement-1 : The variance of first n even natural numbers is n^2-1 4 Statement-2 : The sum of first n natural numbers is n(n+1) 2 and the sum of squares of first n natural numbers is n(n+1)(2 n+1) 6

Step-by-step solution

Statement-2 is true Statement-1: Sum of n even natural numbers =n(n+1) aligned & Mean ( x )= n(n+1) n =n+1 & Variance = [ 1 n (x₁ )^2 ]-( x )^2= 1 n [2^2+4^2+ .+(2 n)^2 ]-(n+1)^2 & = 1 n 2^2 [1^2+2^2+ .+n^2 ]-(n+1)^2= 4 n n(n+1)(2 n+1) 6 -(n+1)^2 & = (n+1)[2(2 n+1)-3(n+1)] 3 = (n+1)[4 n+2-3 n-3] 3 = (n+1)(n-1) 3 = n^2-1 3 aligned Statement 1 is false.