If a 1 ( > 0 ) , a 2 , a 3 , a 4 , a 5 are in a G.P. , a 2 + a 4 = 2 a 3 + 1 and 3 a 2 + a 3 = 2 a 4 , then a 2 + a 4 + 2 a 5 is equal to _____.

Step-by-step solution

Given, a 1 > 0 , a 2 , a 3 , a 4 , a 5 → G.P. 3 a 2 + a 3 = 2 a 4 Using the formula a n = a r n - 1 and assuming a 1 = a we get, 3 a r + a r 2 = 2 a r 3 ⇒ 3 + r = 2 r 2 ⇒ 2 r 2 - r - 3 = 0 ⇒ r = - 1 & r = 3 2 Also, a 2 + a 4 = 2 a 3 + 1 ⇒ a r + a r 3 = 2 a r 2 + 1 ⇒ a r + r 3 - 2 r 2 = 1 ⇒ a 3 2 + 27 8 - 18 4 = 1 ⇒ a = 8 3 When r = - 1 , a = - 1 4 (rejected, a 1 > 0 ) So, at r = 2 3 ,   a = 8 3 Now a 2 + a 4 + 2 a 5 = 8 3 × 3 2 + 8 3 × 27 8 + 2