MyQuestionIcon

1

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

Fractional Exponent

If A 1 A 2 ...

Question

If $A_{1} A_{2}; G_{1} G_{2}; H_{1} H_{2}$ be two A.M.s, G.M.s and H.M.s between two numbers then prove: $\frac{G_{1} G_{2}}{H_{1} H_{2}} = \frac{A_{1} + A_{2}}{H_{1} + H_{2}}$

Open in App

Solution

Let two nos be $a$ and $b$

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

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

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

$a, C_{r_{1}}, b, C_{r_{2}}$ are in G.P

$\sqrt{a b} = \sqrt{C_{r_{1}} C_{r_{2}}}$

$a b = C_{r_{1}} C_{r_{2}}$

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

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

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

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

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

$\frac{A_{1} + A_{2}}{H_{1} + H_{2}} * \frac{H_{1} H_{2}}{C_{r_{1}} C_{r_{2}}} = a + b * \frac{a b}{a + b} * \frac{1}{a b}$

Suggest Corrections

thumbs-up

0

Similar questions

A_{1}, A_{2}; G_{1}, G_{2}

H_{1}, H_{2}

be two A.M.s, G.M.s and H.M.s between two quantities a and b then
prove that

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

A_{1}, A_{2}; G_{1}, G_{2}

H_{1}, H_{2}

be two A.M.s, G.M.s and H.M.s between two quantities a and b then
prove that

A_{1} H_{2} = A_{2} H_{1} = G_{1} G_{2} = a b

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

H_{1}, H_{2}

are two A.Ms, G.Ms and H.M.s between two numbers, then

\frac{A_{1} + A_{2}}{H_{1} + H_{2}} . \frac{H_{1} H_{2}}{G_{1} G_{2}}

A_{1}, A_{2}; G_{1}, G_{2}; H_{1}, H_{2}

A . M .^{'} s, G . M .^{'} s

H . M .^{'} s

between two positive numbers then prove:

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

A_{1}, A_{2}; G_{1}, G_{2}

H_{1}, H_{2}

be AM's, GM's and HM's between two quantities, then the value of

\frac{G_{1} G_{2}}{H_{1} H_{2}}

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Laws of Indices

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Fractional Exponent

Standard IX Mathematics

Join BYJU'S Learning Program

CrossIcon