There is n arithmetic means between 9 and 27. If the ratio of the last mean to the first mean is 2 : 1, find the value of n.

To find: The value of n

Given: (i) The numbers are 9 and 27

(ii) The ratio of the last mean to the first mean is 2 : 1

Formula used: (i) , where, d is the common difference

n is the number of arithmetic means

(ii) A_n = a + nd

We have 9 and 27,

Using Formula,

Using Formula, A_n = a + nd

First mean i.e.,

… (i)

Last mean i.e.,

A_n … (ii)

The ratio of the last mean to the first mean is 2 : 1

Substituting the value of A₁ and A_n from eqn. (i) and (ii)

⇒ 27n + 9 = 18n + 54

⇒ 9n = 45

⇒ n = 5

Ans) The value of n is 5