The first three terms of an arithmetic-geometric progression are 3 , – 1 and – 1 . The next term of the progression is

Step-by-step solution

Let the first four terms of the A . G . P . be 3 ,   3 + d r ,   3 + 2 d r 2 ,   3 + 3 d r 3 Hence, 3 + d r = - 1 and 3 + 2 d r 2 = - 1 3 + d 2 r 2 3 + 2 d r 2 = 1 - 1 ⇒ 9 + d 2 + 6 d = - 3 - 2 d ⇒ d 2 + 8 d + 12 = 0 ⇒ d = - 2 , - 6 r = - 1 3 + d ⇒ r = - 1 or 1 3 (i) for d = - 2 ,   r = - 1 ; next term is 3 + 3 d r 3 = 3 - 6 - 1 3 = 3 (ii) for d = - 6 , r = 1 3 ; next term is 3 + 3 d r 3 = 3 - 18 1 3 3 = - 15 27 = - 5 9 So, the next term is 3 or - 5 9