If n arithmetic means are inserted between 50 and 200, and n harmonic

1.	If n arithmetic means are inserted between 50 and 200, and n harmonic means are inserted between the same two numbers, then a2 .hn–1 is equal to(a) 500(b) 5000(c) 10,000(d) None of these
Answer» Correct option (c) 10,000 Explanation: 50, a₁ , a₂ ,...........a_n , 200 are in AP ......... (1) Also, 50, h₁ , h₂ ,.........,h_n 200 are in H.P ⇒ 1/50, 1/h₁, 1/h₂ , .... 1/h_n, 1/200 are in AP ⇒ 1/200, 1/h₁, 1/h_{n - 1}, ....1/h₁, 1/50 are in AP Multiply by 200×50 = 10,000 ⇒ 50, 10,000/h_n , 10,000/h_{n - 1}, ....10,000/h₂, 10,000/h₁, 200 are in AP ... (2) Now (1) and (2) are identical. ⇒ a₂ = 10,000/h_{n - 1} gives a₂ .h_{n – 1} = 10,000

If n arithmetic means are inserted between 50 and 200, and n harmonic means are inserted between the same two numbers, then a2 .hn–1 is equal to(a) 500(b) 5000(c) 10,000(d) None of these

Answer»

Correct option (c) 10,000

Explanation:

50, a₁ , a₂ ,...........a_n , 200 are in AP ......... (1)

Also, 50, h₁ , h₂ ,.........,h_n 200 are in H.P

⇒ 1/50, 1/h₁, 1/h₂ , .... 1/h_n, 1/200 are in AP

⇒ 1/200, 1/h₁, 1/h_{n - 1}, ....1/h₁, 1/50 are in AP

Multiply by 200×50 = 10,000

⇒ 50, 10,000/h_n , 10,000/h_{n - 1}, ....10,000/h₂, 10,000/h₁, 200 are in AP ... (2)

Now (1) and (2) are identical.

⇒ a₂ = 10,000/h_{n - 1} gives a₂ .h_{n – 1} = 10,000