Organ pipes of lengths $l_{1}, l_{2}, l_{3}$ emit notes of frequency $n_{1}, n_{2}, n_{3}$ . lf $l_{1}, l_{2}, l_{3}$ are in A.P, then n $_{2}$ is

\frac{2 n_{1} n_{3}}{n_{1} + n_{3}}

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\frac{n_{1} + n_{3}}{2}

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\sqrt{n_{1} n_{3}}

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\sqrt{\frac{n_{1}^{2} n_{3}^{2}}{2}}

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Solution

The correct option is A $\frac{2 n_{1} n_{3}}{n_{1} + n_{3}}$
$n_{1} = \frac{ν}{4 L_{1}} \Rightarrow L_{1} = \frac{ν}{4 n_{1}}$

$n_{2} = \frac{ν}{4 L_{1}} \Rightarrow L_{2} = \frac{ν}{4 n_{2}}$

$n_{3} = \frac{ν}{4 L_{3}} \Rightarrow L_{3} = \frac{ν}{4 n_{3}}$

if $L_{1}, L_{2}, L_{3},$ are in A.P.

$L_{3} - L_{2} = L_{2} - L_{1}$

$\frac{ν}{4 n_{3}} - \frac{ν}{4 n_{2}} = \frac{ν}{4 n_{2}} - \frac{ν}{4 n_{1}}$

$\frac{1}{n_{3}} + \frac{1}{n_{1}} = \frac{2}{n_{2}}$

$\Rightarrow n_{2} = \frac{2}{\frac{1}{n_{1}} + \frac{1}{n_{3}}}$

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