Let a , b , c > 1 , a 3 , b 3 and c 3 be in A . P . and log a b , log c a and log b c be in G . P . If the sum of first 20 terms of an A . P . , whose first term is a + 4 b + c 3 and the common difference is a - 8 b + c 10 is - 444 , then a b c is equal to

Step-by-step solution

Given, a 3 , b 3 , c 3 be in A . P . So, a 3 + c 3 = 2 b 3   . . . . . . . . 1 Also given log a b ,   log c a   &   log b c are in G . P . So, log b log a · log c log b = log a log c 2 ⇒ ( log a ) 3 = ( log c ) 3 ⇒ a = c ..........(2) Now from equation 1   &   2 we get, a = b = c Now, T 1 = a + 4 b + c 3 = 2 a and d = a - 8 b + c 10 = - 6 a 10 = - 3 5 a ∴ S 20 = 20 2 4 a + 19 - 3 5 a ⇒ S 20 = 10 20 a - 57 a 5 ⇒ S 20 = - 74 a ⇒ - 444 = -