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A.M Greater than or Equal to G.M

If a, a 1 ,...

Question

If $a, a_{1}, a_{2}, . . . . ., a_{2 n}, b$ 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 $a$ and $b$ , then $\frac{a_{1} + a_{2 n}}{g_{1} g_{2 n}} + \frac{a_{2} + a_{2 n - 1}}{g_{2} g_{2 n - 1}} + . . . . . + \frac{a_{n} + a_{n + 1}}{g_{n} g_{n + 1}}$ is equal to

2 n h

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n / h

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n h

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2 n / h

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Solution

The correct option is D $2 n / h$
Since $a, a_{1}, a_{2}, . . . . a_{2 n}, b$ are in arithmetic progression
$a, a_{1}, a_{2}, . . . . a_{2 n}$ are A.M's between $a$ and $b$
$∴ a_{1} + a_{2 n} = a_{2} + a_{2 n - 1} = . . . . = a + b$
similarly $a, g_{1}, g_{2}, . . . . g_{2 n}$ are in G.P
$∴ g_{1} g_{2 n} = g_{2} g_{2 n - 1} = . . . . = g_{n} g_{n + 1} = a b$
$∴ \frac{a_{1} + a_{2 n}}{g_{1} g_{2 n}} + \frac{a_{2} + a_{2 n - 1}}{g_{2} g_{2 n - 1}} + . . . . \frac{a_{n} + a_{n + 1}}{g_{n} g_{n + 1}}$
$= \frac{a + b}{a b} + \frac{a + b}{a b} + . . . . . . + \frac{a + b}{a b} = \frac{n (a + b)}{a b} = \frac{2 n}{h}$
where $h$ is H.M between $a$ and $b$
$\Rightarrow h = \frac{2 a b}{a + b}$

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