Let A and B be two sets containing the terms of two distinct finite arithmetic progressions with positive common differences. Set A consists of 11 terms with a mean of 20 and a variance of 10 . Set B consists of 15 terms with a mean of 20 and a variance of 168 . The sum of all elements in the set A B is equal to

Step-by-step solution

For an arithmetic progression of n terms with common difference d , the variance is given by ^2 = n^2 - 1 12 d^2 . For set A : n = 11 , ^2 = 10 11^2 - 1 12 d₁^2 = 10 120 12 d₁^2 = 10 10 d₁^2 = 10 d₁ = 1 Since the mean is the middle (6th) term, the terms of A are symmetric around 20 . A = 20 - 5(1), , 20 + 5(1) = 15, 16, , 25 Sum of elements in A = 11 20 = 220 . For set B : n = 15 , ^2 = 168 15^2 - 1 12 d₂^2 = 168 224 12 d₂^2 = 168 56 3 d₂^2 = 168 d₂^2 = 9 d₂ = 3 The mean is the middle (8th) term, so the terms of B