Let a 1 , a 2 , a 3 , … be an A.P. If ∑ r = 1 ∞ a r 2 r = 4 , then 4 a 2 is equal to ______.

Step-by-step solution

Let S = ∑ r = 1 ∞ a r 2 r ⇒ S = a 1 2 + a 2 2 2 + a 3 2 3 + …     . . . . . . 1 ⇒ S 2 =    a 1 2 2 + a 2 2 3 + …     . . . . . 2 Now subtracting both equation we get, ⇒ S 2 = a 1 2 + d 1 2 2 + 1 2 3 + … ⇒ S 2 = a 1 2 + d 1 4 1 - 1 2 ⇒ S = a 1 + d = a 2 Now given S = ∑ r = 1 ∞ a r 2 r = 4 So, S = a 2 = 4 So, 4 a 2 = 16