If each term of a geometric progression a 1 , a 2 , a 3 , … with a 1 = 1 8 and a 2 ≠ a 1 , is the arithmetic mean of the next two terms and S n = a 1 + a 2 + … + a n , then S 20 - S 18 is equal to

Step-by-step solution

Let GP : a , ar , ar 2 , ar 3 , . . . Now, given: 2 ar = ar 2 + ar 3 ⇒ 2 = r + r 2 ⇒ r 2 + r - 2 = 0 ⇒ r + 2 r - 1 = 0 ⇒ r = - 2 , 1 Since GP is not constant so r = 1 is rejected ⇒ r = - 2 ⇒ S 20 - S 18 = a 1 - r 20 1 - r - a 1 - r 18 1 - r ⇒ S 20 - S 18 = a r 18 - r 20 1 - r ⇒ S 20 - S 18 = 2 18 - 2 20 8 × 3 ⇒ S 20 - S 18 = 2 18 1 - 4 24 ⇒ S 20 - S 18 = - 2 15