Question

Insert arithmetic means between 16 and 65 such that the 5^th AM is 51. Find the number of arithmetic means.

Answer 1

To find: The number of arithmetic means

Given: (i) The numbers are 16 and 65

(ii) 5^th arithmetic mean is 51

Formula used: (i) d = b-a/n+1, where, d is the common difference

n is the number of arithmetic means

(ii) A_n = a + nd

We have 16 and 65,

d = 7

Using Formula, A_n = a + nd

First arithmetic mean, A₁ = a + d

= 16 + 7 = 23

Second arithmetic mean, A₂ = a + 2d

= 16 + 2(7) = 16 + 14 = 30

Third arithmetic mean, A₃ = a + 3d

= 16 + 3(7) = 16 + 21 = 37

Fourth arithmetic mean, A₄ = a + 4d

= 16 + 4(7) = 16 + 28 = 44

Fifth arithmetic mean, A₅ = a + 5d

= 16 + 5(7) = 16 + 35 = 51

Sixth arithmetic mean, A₆ = a + 6d

= 16 + 6(7) = 16 + 42 = 58

(Ans) The six arithmetic means between 1 and 65 are 23, 30, 37, 44, 51 and 58.