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

Question

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, 4 l ⁡ 2 m 2 n 2, 4 l ⁡ 2 m n, 4 l ⁡ m 2 n, 4 l ⁡ m n 2

MARKS App · Accepted Answer

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 + 2 r 4 + r 8 = l 4 × n l ⁡ 1 + r 4 2 = l ⁡ 3 n × 1 + n l ⁡ 2 = l ⁡ 3 × n × l ⁡ + n 2 l ⁡ 2 = l ⁡ n × 2 m 2 (From (1)) = 4 l ⁡ m 2 n

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