Question

Find :

(i) 4 H.M. between 1 and 1/16.

(ii) 5 H.M. between 1/19 and 1/7.

(iii) 1 H.M. between – 2/5 and 4/25.

Answer 1

(i) Let H₁, H₂, H₃, H₄ are 4 H.M. between 1 and 1/16.

So, 1, H₁, H₂, H₃, H₄, 1/16 are in H.P.

First term of corresponding A.P. is 1 and 6^th term is 16.

∴ a + 5d = 16 (∵ T₆ = a + 5d)

⇒ 1 + 5d = 16

⇒ 5d = 15 ⇒ d = 3 (∵ a²)

So. there will be 4 AM. between 1 and 16.

1 + d, 1 + 2d, 1 + 3d, 1 + 4d

or 1 + 3, 1 + 2(3), 1 + 3(3), 1 + 4(3)

or 4, 7, 10, 13

Hence, required H.M.will be 1/4, 1/7, 1/10, 1/13.

(ii) 5 H.M. between 1/19 and 1/7.

So, corresponding A.M. are a + d, a + 2d, a + 3d, a + 4d, a + 5d

or 19 – 2, 19 + 2(- 2), 19 + 3(- 2), 19 + 4(- 2), 19 + 5(- 2)

or 17, 15, 13, 11,9

Hence, corresponding H.M. will be

1/17, 1/15, 1/13, 1/11, 1/9

(iii) 1 H.M. between – 2/5 and 4/25.