If a, b, c are in H.P.,then which one of the | vedclass

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(D) Given that $a, b, c$ are in $H.P.$,the condition for harmonic progression is $b = \frac{2ac}{a + c}$.Let us check option $(a)$:$\frac{1}{b - a} +

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None of these

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(D) Given that $a, b, c$ are in $H.P.$,the condition for harmonic progression is $b = \frac{2ac}{a + c}$.Let us check option $(a)$:$\frac{1}{b - a} + \frac{1}{b - c} = \frac{b - c + b - a}{(b - a)(b - c)} = \frac{2b - (a + c)}{b^2 - b(a + c) + ac}$.Substituting $b = \frac{2ac}{a + c}$,we find this does not equal $\frac{1}{b}$.Let us check option $(b)$:$\frac{ac}{a + c} = b$ $\Rightarrow \frac{ac}{a + c} = \frac{2ac}{a + c}$ $\Rightarrow 1 = 2$,which is false.Let us check option $(c)$:$\frac{b + a}{b - a} + \frac{b + c}{b - c} = \frac{(b + a)(b - c) + (b + c)(b - a)}{(b - a)(b - c)} = \frac{2b^2 - 2ac}{b^2 - b(a + c) + ac}$.Substituting $b = \frac{2ac}{a + c}$,we find this does not equal $1$.Therefore,none of the given options are true.

If $a, b, c$ are in $H.P.$,then which one of the following is true?

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