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

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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 $A_{1}, A_{2}, . . ., A_{n} a n d H_{1}, H_{2}, . . ., H_{n}$

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