Question

If $n$ arithmetic means $a_1,a_2,...,a_n$ are inserted between $50$ and $200$ and $n$ harmonic means $h_1,h_2,...,h_n$ are inserted between the same two numbers, then $(a_2{h}_{n-1}÷2000)$ is equal to

Answer 1

$50,a_1,a_2,...,a_n,200$ are in $A.P.,$ ...(1)

And

$50,h_1,h_2,...,h_n,200$ are in $H.P.,$

$\Rightarrow \frac{1}{50},\frac{1}{h_1},\frac{1}{h_2},...,\frac{1}{h_n},\frac{1}{200}$ are in $A.P.$

Reversing it, we get

$\frac{1}{200},\frac{1}{h_n},\frac{1}{h_{n-1}},...,\frac{1}{h_1},\frac{1}{50}$ are in $A.P.$ ...(2)

Multiplying (2) by $10000$, we get

$50,\frac{10000}{h_n},\frac{10000}{h_{n-1}},...,\frac{10000}{h_1},200$ are in $A.P.$ ... (3)

(1) and (3) are identical,

$\therefore a_2=\frac{10000}{h_{n-1}}\Rightarrow a_2{h}_{n-1}=10000$

Hence, answer is $10000$.

So, $a_2{h}_{n-1}÷2000=5$