You visited us 1 times! Enjoying our articles? Unlock Full Access!

Harmonic Mean

Two A.M's A...

Question

Two A.M's $A_{1}$ and $A_{2}$ two G.M's G1,G2 are inserted between any two numbers, then $[{H_{1}}^{- 1} + {H_{2}}^{- 1}]$ equals: (where H stands for harmonic mean) inserted between given two numbers.

Open in App

Solution

Given,

Let the 2 numbers be $a, b$

As $a, A_{1}, A_{2}, b$ are in A.P

$\frac{a + b}{2} = \frac{A_{1} + A_{2}}{2}$

$∴ a + b = A_{1} + A_{2}$ ......(1)

As $a, G_{1}, G_{2}, b$ are in G.P

$\sqrt{a b} = \sqrt{G_{1} G_{2}}$

$∴ a b = G_{1} G_{2}$ .......(2)

As $a, H_{1}, H_{2}, b$ are in H.P

$\Rightarrow \frac{1}{a}, \frac{1}{H_{1}}, \frac{1}{H_{2}}, \frac{1}{b}$ are in A.P

$\frac{\frac{1}{a} + \frac{1}{b}}{2} = \frac{\frac{1}{H_{1}} + \frac{1}{H_{2}}}{2}$

$\frac{a + b}{a b} = \frac{H_{1} + H_{2}}{H_{1} H_{2}}$

$∴ \frac{1}{H_{1}} + \frac{1}{H_{2}} = \frac{a + b}{a b}$
$∴ \frac{1}{H_{1}} + \frac{1}{H_{2}} = \frac{A 1 + A 2}{G 1 G 2}$

Suggest Corrections

Similar questions

A_{1}

A_{2}

G_{1}, G_{2}

H_{1}, H_{2}

\frac{1}{H_{1}} + \frac{1}{H_{2}} =

A_{1} a n d A_{2}

G_{1} a n d G_{2}

H_{1} a n d H_{2}

H_{1} a n d H_{2}

A_{1}, A_{2}, G_{1}, G_{2} .

Two AMs $A_{1}$ and $A_{2}$ , two GMs $G_{1}$ and $G_{2}$ and two HMs $H_{1}$ and $H_{2}$ are inserted between two numbers a and b, then

$\frac{1}{H_{1}}$ + $\frac{1}{H_{2}}$ equals.

A_{1}, A_{2}

G_{1}, G_{2} : H_{1} H_{2} = A_{1} + A_{2} : H_{1} + H_{2}

n

a_{1}, a_{2}, . . ., a_{n}

50

200

n

h_{1}, h_{2}, . . ., h_{n}

(a_{2} h_{n - 1} \div 2000)

Join BYJU'S Learning Program