Let A 1 , A 2 , A 3 be the three A.P. with the same common difference d and having their first terms as A , A + 1 , A + 2 , respectively. Let a , b , c be the 7 th , 9 th , 17 th terms of A 1 , A 2 , A 3 , respectively such that a 7 1 2 b 17 1 c 17 1 + 70 = 0 . If a = 29 , then the sum of first 20 terms of an AP whose first term is c - a - b and common difference is d 12 , is equal to _____ .

Step-by-step solution

From the definition of A.P., we have a = A + 6 d       . . . . i b = A + 1 + 8 d       . . . . ii c = A + 2 + 16 d       . . . . iii Now, we are given a 7 1 2 b 17 1 c 17 1 + 70 = 0 ⇒ A + 6 d 7 1 2 A + 2 + 16 d 17 1 A + 2 + 16 d 17 1 + 70 = 0 ⇒ A + 6 d 7 1 2 + 4 d 3 - 1 - A 0 0 + 70 = 0 ⇒ 10 A + 70 = 0 ⇒ A = - 7 So, a = A + 6 d ⇒ 29 = - 7 + 6 d ⇒ d = 6 ∴ c - a - b = 1 + 2 d - A = 13 + 7 = 20 And, d 12 = 6 12 = 1 2 So, S 20 = 20 2