Question

If $n$ arithmetic means $a_1, a_2, \ldots . ., a_n$ are inserted between 50 and 200 and $n$ harmonic means $h_1, h_2, \ldots .$. $h_n$ 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)

$50, a _1, a _2, \ldots \ldots . . . ., a _{ n }, 200$ are in A.P. ....(1)
$50, h_1, h_2, \ldots . . . . . ., h_n, 200$ are in H.P.
$\frac{1}{50}, \frac{1}{ h _1}, \frac{1}{ h _2}, \ldots \ldots ., \frac{1}{ h _{ n }}, \frac{1}{200}$ are in A.P.
Reversing,
$\frac{1}{200}, \frac{1}{ h _{ n }}, \frac{1}{ h _{ n -1}}, \ldots \ldots . ., \frac{1}{ h _1}, \frac{1}{50}$ are in A.P.
Multiplying by 10000 , we get
$50, \frac{10000}{ h _{ n }}, \frac{10000}{ h _{ n -1}}, \ldots \ldots . ., \frac{10000}{ h _1}, 200$ are in A.P. ......(2)
(1) and (2) are identical. $ \therefore a _2=\frac{10000}{ h _{ n -1}}$
$a _2 h _{ n -1}=10000 $