Question

If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4^th A.M. is equal to 2^nd G.M., then m is equal to ______.

विकल्प

• 36

• 37

• 38

• 39

Answer 1

If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4^th A.M. is equal to 2^nd G.M., then m is equal to 39.

Explanation:

Let m arithmetic mean be A₁, A₂ ... A_m and G₁, G₂, G₃ be geometric mean.

The A.P. formed by arithmetic mean is 

3, A₁, A₂, A₃, ....... A_m, 243

∴ d = `(243 - 3)/("m" + 1) = 240/("m" + 1)`

The G.P. formed by geometric mean 3, G₁, G₂, G₃, 243

∴ r = `((243)/3)^(1/(3 + 1)) = (81)^(1/4)` = 3 ∵ A₄ = G₂

⇒ `3 + 4(240/("m" + 1))` = 3(3)²

⇒ `3 + 960/("m" + 1)` = 27

⇒ m + 1 = 40

⇒ m = 39.