Three positive numbers form a geometric progression and their sum is 14 . If 1 is added to the first number, 2 to the second number, and 1 to the third number, the resulting three numbers form an arithmetic progression. The sum of the squares of the original three numbers is _____.

Step-by-step solution

Let the three positive numbers in G.P. be a, ar, ar^2 . Given that their sum is 14 : a(1 + r + r^2) = 14 ... (1) When 1, 2, and 1 are added to these numbers respectively, the new numbers a+1, ar+2, ar^2+1 form an A.P. Using the condition for an A.P. ( 2b = a + c ): 2(ar + 2) = (a + 1) + (ar^2 + 1) 2ar + 4 = a + ar^2 + 2 a(r^2 - 2r + 1) = 2 a(r - 1)^2 = 2 ... (2) Dividing equation (1) by equation (2): 1 + r + r^2 r^2 - 2r + 1 = 7 1 + r + r^2 = 7r^2 - 14r + 7 6r^2 - 15r + 6 = 0 2r^2 - 5r + 2 = 0 (2r - 1)(r - 2) = 0 T