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) , where, d is the common difference

n is the number of arithmetic means

(ii) A_n = a + nd

We have 16 and 65,

Using Formula,

Using Formula, A_n = a + nd

Fifth arithmetic mean, A₅ = a + 5d

A₅ = 51 (Given)

Therefore,

⇒ 245 = 35n + 35

⇒ 210 = 35n

⇒ n = 6

The number of arithmetic means are 6.

Using Formula,

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.