If the harmonic mean between a and b be H , then the value of H − a 1 + H − b 1 is

Putting H = a + b 2 ab , we have H − a 1 + H − b 1 = ( a + b 2 ab − a ) 1 + ( a + b 2 ab − b ) 1 = ab − a 2 a + b + ab − b 2 a + b = ( b − a a + b ) ( a 1 − b 1 ) = ( b − a a + b ) ( ab b − a ) = ab a + b = a 1 + b 1

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MathematicsSequence & SeriesA.P.Easy2 minPYQ_2021

MathematicsEasysingle choice

If the harmonic mean between $a$ and $b$ be $H$ , then the value of $\frac{1}{H - a} + \frac{1}{H - b}$ is

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Solution:

Putting $H = \frac{2 ab}{a + b}$ , we have $\frac{1}{H - a} + \frac{1}{H - b} = \frac{1}{( \frac{2 ab}{a + b} - a )} + \frac{1}{( \frac{2 ab}{a + b} - b )} = \frac{a + b}{ab - a ^{2}} + \frac{a + b}{ab - b ^{2}}$ $= (\frac{a + b}{b - a}) (\frac{1}{a} - \frac{1}{b}) = (\frac{a + b}{b - a}) (\frac{b - a}{ab}) = \frac{a + b}{ab} = \frac{1}{a} + \frac{1}{b}$

Stream:BITSATSubject:MathematicsTopic:Sequence & SeriesSubtopic:A.P.

⏱ 2mℹ️ Source: PYQ_2021

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