Let the positive numbers a 1 , a 2 , a 3 , a 4 and a 5 be in a G.P. Let their mean and variance be 31 10 and m n respectively, where m and n are co-prime. If the mean of their reciprocals is 31 10 and a 3 + a 4 + a 5 = 14 , then m + n is equal to ____________.

Step-by-step solution

5 positive numbers a 1 ,   a 2 , . . . ,   a 5 are in geometric progression. So, let the numbers be a r 2 ,   a r ,   a ,   a r ,   a r 2 According to question, a r 2 + a r + a + a r + a r 2 5 = 31 10       . . . 1 And, r 2 a + r a + 1 a + 1 a r + 1 a r 2 5 = 31 40 ⇒ 1 a r 2 + r + 1 + 1 r + 1 r 2 5 = 31 40       . . . 2 By 1   &   2 , we get a 2 = 4   ⇒ a = 2 So, r 2 + r + 1 + 1 r + 1 r 2 = 31 4 ⇒ r + 1 r 2 + r + 1 r =