If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4 t h A.M. is equal to 2 n d G.M., then m is equal to:

Step-by-step solution

Let 3 ,   A 1 ,   A 2 ,   A 3 , . . . . . . A m ,   243 are in arithmetic progression with m arithmetic means. Common difference d = 243 - 3 m + 1 = 240 m + 1 Let 3 ,   G 1 ,   G 2 ,   G 3 ,   243 are in geometric progression with 3 geometric means. Common ratio r = 243 3 1 3 + 1 = 81 1 4 = 3 Given G 2 = A 4 ⇒ 3 3 2 = 3 + 4 240 m + 1 ⇒ 27 = 3 + 960 m + 1 ⇒ m + 1 = 40 ⇒ m = 39