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If m is the A . M . of two distinct real numbers I and n I , n > 1 and G 1 , G 2 and G 3 are three geometric means between I and n , then G 1 4 + 2 G 2 4 + G 3 4 equals

Options

  1. A4 l ⁡ 2 m 2   n 2
  2. B4   l ⁡ 2 m n
  3. C4   l ⁡ m 2   n
  4. D4 l ⁡ m n 2

Correct answer

C. 4   l ⁡ m 2   n

Step-by-step solution

Given, m is the A.M. of l   &   n .   ⇒ m = l + n 2     . . . 1 G 1 ,   G 2 ,   G 3 are G.M. between l   &   n . ⇒ l ,   G 1 ,   G 2 ,   G 3 ,   n are in G.P. ⇒ n = l r 4   ⇒ r 4 = n l     . . . 2 ⇒ G 1 = l ⁡ r ,   G 2 = l ⁡ r 2 ,   G 3 = l ⁡ r 3 ⇒ G 1 4 + 2 G 2 4 + G 3 4 = l ⁡ 4 r 4 + 2 × l ⁡ 4 r 8 + l ⁡ 4 × r 12 = l ⁡ 4 r 4 1

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