CameraIcon

SearchIcon

MyQuestionIcon

1

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

Quantitative Aptitude

A.M Greater than or Equal to G.M

Let a1, a2,...

Question

Let $a_{1}, a_{2}, . . . .$ 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 $G_{1}, G_{2}, . . . ., G_{n}$ in terms of $A_{1}, A_{2}, . . . . ., A_{n}, H_{1}, H_{2}, . . . . ., H_{n} .$

Open in App

Solution

$a_{1}, a_{2}, . . . .,$ are in G.P.
$∴ a_{2} = a_{1} r, a_{3} = a_{1} r^{2}, . . . ., a_{n} = a_{1} r^{n - 1}$
$A_{n} = \frac{\sum a_{1}}{n} = \frac{a_{1}}{n} (1 + r + r^{2} + . . . r^{n - 1})$
or $A_{n} = \frac{a_{1}}{n} (\frac{r^{n} - 1}{r - 1})$
$G_{n} = (a_{1} a_{2} . . . a_{n})^{1 / n} = a_{1} (1. r . r^{2} . . . . r^{1 - n})^{1 / n}$
or $G_{n} = a_{1} [r^{(1 / 2) (n - 1)}]^{1 / n} = a_{1} r^{n - 1})$
$\frac{1}{H_{n}} = \frac{1}{n} \sum \frac{1}{a_{1}} = \frac{1}{n} . \frac{1}{a_{1}} [1 + \frac{1}{r} + \frac{1}{r^{2}} + . . . \frac{1}{r^{n - 1}}]$
$= \frac{1}{n . a_{1}} . \frac{1 - (1 / r)^{n}}{1 - 1 / r} = \frac{1}{n . a_{1}} . \frac{r^{n} - 1}{r - 1} . \frac{1}{r^{n - 1}}$
$∴ H_{n} = \frac{n a_{1} r^{n - 1} (r - 1))}{r^{n} - 1}$
From (2), (3) and (4) we observe that
$G_{n}^{2} = A_{n} H_{n} = a_{1}^{2} r^{(n - 1)}$
Above is true for each n, hence
$G_{n}^{2} G_{2}^{2} . . . G_{n}^{2} = (A_{1} A_{2} . . . A_{n}) (H_{1} H_{2} . . . H_{n})$
$∴ (G_{1} G_{2} . . . . G_{n})^{1 / n} = [(A_{1} A_{2} . . . . A_{n}) (H_{1} H_{2} . . . . H_{n})]^{1 / 2 n}$
L.H.S. is G.M. of $G_{1}, G_{2}, . . ., G_{n}$ whose value we have determined in terms of $A_{1}, A_{2}, . . ., A_{n} a n d H_{1}, H_{2}, . . ., H_{n}$

Suggest Corrections

thumbs-up

0

Similar questions

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

be positive real numbers in geometric progression . For each

n

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

be respectively, the A.M, G.M and H.M of

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

. Find an expression for the G.M of

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

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

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

A_{1}, G_{1}

H_{1}

denote the arithmetic, geometric and harmonic means respectively of two distinct positive numbers. For

n \geq 2

A_{n - 1}

H_{n - 1}

have arithmetic, geometric and harmonic means as

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

respectively.
Which of the following statements is correct?

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_{1}, G_{1}, H_{1}

denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For

n \geq 2,

A_{n - 1}

H_{n - 1}

hav arithmetic, gemetric and hrmonic means as

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

Which one of the following statements is correct?

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

are in arithmetic progression and

a, g_{1}, g_{2}, . . . . g_{2 n}, b

are in geometric progression and

h

is the harmonic mean of

a

b

\frac{a_{1} + a_{2 n}}{g_{1} g_{2 n}} + \frac{a_{2} + a_{2 n - 1}}{g_{2} g_{2 n - 1}} + . . . . . + \frac{a_{n} + a_{n + 1}}{g_{n} g_{n + 1}}

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Inequations I

QUANTITATIVE APTITUDE

Watch in App

explore_more_icon

Explore more

A.M Greater than or Equal to G.M

Other Quantitative Aptitude

Join BYJU'S Learning Program

CrossIcon