MyQuestionIcon

1

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

Mean

Let a, b be...

Question

Let $a, b$ be positive real numbers. If $a, G_{1}, G_{2}, b$ are in geometric progression and $a, H_{1}, H_{2}, b$ are in harmonic progression, show that
$\frac{G_{1} G_{2}}{H_{1} H_{2}} = = \frac{(2 a + b) (a + 2 b)}{9 a b}$

Open in App

Solution

Now 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.
$∴ \frac{1}{b} = \frac{1}{a} + 3 D$ $∴ D = \frac{1}{3} (\frac{1}{b} - \frac{1}{a})$
$∴ \frac{1}{H_{1}} = \frac{1}{a} + D = \frac{1}{a} + \frac{1}{3} (\frac{1}{b} - \frac{1}{a}) = \frac{2 b + a}{3 a b}$
$∴ \frac{1}{H_{2}} = \frac{1}{a} + 2 D = \frac{1}{a} + \frac{2}{3} (\frac{1}{b} - \frac{1}{a}) = \frac{2 a + b}{3 a b}$
$∴ \frac{G_{1} G_{2}}{H_{1} H_{2}} = \frac{a b (2 b + a) (2 a + b)}{9 a^{2} b^{2}} = \frac{(2 b + a) (2 a + b)}{9 a b}$
Note : The value of $H_{2}$ can be obtained from the value of $H_{1}$ if we interchange a and b.

Suggest Corrections

thumbs-up

0

Similar questions

A_{1}

A_{2}

be two arithmetic means ,

G_{1}

G_{2}

be two geometric means and

H_{1}

H_{2}

be two harmonic means between two positive numbers a and b . The value of

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

a

b

be two positive real numbers. Suppose

A_{1}, A_{2}

are two arithmetic means;

G_{1}, G_{2}

are two geometric means and

H_{1}, H_{2}

are two harmonic means between

a

b

A_{1}, A_{2};

G_{1}, G_{2}

H_{1}, H_{2}

are arithmetic mean, geometric mean and harmonic mean between two numbers, then the value of

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

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

be positive real numbers in geometric progression. For each n, let

A_{n}, G_{n}, H_{a}

be respectively, the arithmetic mean, geometric mean and harmonic mean of

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

Find an expression for the geometric mean of

G_{1}, G_{2}, . . . ., G_{n}

A_{1}, A_{2}, . . . . ., A_{n}, H_{1}, H_{2}, . . . . ., H_{n} .

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}}

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Central Tendency

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon