Question

If $n$ arithmetic means $a_1,a_2,\dots ..,a_n$ are inserted between 50 and 200 and $n$ harmonic means $h_1,h_2,\dots .$. $h_n$ are inserted between the same two numbers, then $a_2{h}_{n-1}$ is equal to

• A. 500
• B. $\frac{10000}{n}$
• C. 10000
• D. $\frac{250}{n}$

Answer 1

Answer: (C)

$50,a_1,a_2,\dots \dots ....,a_n,200$ are in A.P. ....(1)
$50,h_1,h_2,\dots ......,h_n,200$ are in H.P.
$\frac{1}{50},\frac{1}{h_1},\frac{1}{h_2},\dots \dots .,\frac{1}{h_n},\frac{1}{200}$ are in A.P.
Reversing,
$\frac{1}{200},\frac{1}{h_n},\frac{1}{h_{n-1}},\dots \dots ..,\frac{1}{h_1},\frac{1}{50}$ are in A.P.
Multiplying by 10000 , we get
$50,\frac{10000}{h_n},\frac{10000}{h_{n-1}},\dots \dots ..,\frac{10000}{h_1},200$ are in A.P. ......(2)
(1) and (2) are identical. $\therefore a_2=\frac{10000}{h_{n-1}}$
$a_2{h}_{n-1}=10000$