Question

m A.M and 3 G M are inserted between 3 and 243 such that 2nd GM = 4th AM then m = ?​

Answer 1

There are m A.M and 3 G.M are inserted between 3 and 243 such that 2nd G.M = 4th AM.

To find : The value of m.

solution : given 3, A₁, A₂, A₃, A₄ .... Am, 243

so, d = (b - a)/(n + 1)

here b = 243, a = 3 , n = m

so, d = (243 - 3)/(m + 1) = 240/(m + 1)

now using formula, kth AM = a + kd

so, 4th AM = 3 + 4(240)/(m + 1) = 3 + 960/(m + 1).....(1)

again, 3, G₁ , G₂ , G₃ , 243

so, r = (b/a)^1/(n+1)

here b = 243 , a = 3 , n = 3

now r = (243/3)¼ = 81¼ = 3

using formula, kth GM = arⁿ¯¹

so, 3rd GM = (3)(3)³¯¹ = 27 ......(2)

a/c to question,

3rd GM = 4th AM

⇒27 = 3 + 960/(m + 1)

⇒m = 39

Therefore the value of m is 39.