Let A 1 and A 2 be two arithmetic means and G 1 , G 2 and G 3 be three geometric means of two distinct positive numbers. Then G 1 4 + G 2 4 + G 3 4 + G 1 2 G 3 2 is equal to

Step-by-step solution

Let the two numbers are a ,   b ⇒ a , A 1 , A 2 , b are in AP. ⇒ b = a + 4 - 1 d ⇒ d = b - a 3 ⇒ A 1 = a + b - a 3 = 2 a + b 3 ⇒ A 2 = a + b - a 3 · 2 = a + 2 b 3 Similarly a , G 1 , G 2 , G 3 , b are in GP. ⇒ b = a r 5 - 1 ⇒ r = b a 1 4 ⇒ G 1 = a b a 1 4 ⇒ G 2 = a b a 2 4 ⇒ G 3 = a b a 3 4 = G 1 4 + G 2 4 + G 3 4 + G 1 2 · G 3 2 ⇒ a 4 · b a + a 4 · b 2 a 2 + a 4 · b 3 a 3 + a 4 · b 2 a 2 = b a 3 + b 2 a 2 + b 3 a +