The sum to infinite terms of the arithmetic-geometric progression 3,   4 ,   4 ,   32 9 ,   . . . . is equal to

Step-by-step solution

Let the A.G.P. be 3 ,   3 + d r ,   3 + 2 d r 2 . . . . So, 3 + d r = 4 and 3 + 2 d r 2 = 4 3 + d 2 r 2 3 + 2 d r 2 = 16 4 ⇒ 9 + d 2 + 6 d = 12 + 8 d ⇒ d 2 - 2 d - 3 = 0 ⇒ d + 1 d - 3 = 0 ⇒ d = - 1 , 3 r = 4 3 + d ⇒ r = 4 3 - 1 , 4 3 + 3 = 2 , 2 3 But r < 1 ⇒ d = 3 and r = 2 3 S ∞ = a 1 - r + d r 1 - r 2 = 3 1 - 2 3 + 3 × 2 3 1 - 2 3 2 = 9 + 18 = 27