If a, b, c, d are in H.P.,then | vedclass

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(C) If $a, b, c, d$ are in $H.P.$,then their reciprocals $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \frac{1}{d}$ are in $A.P.$For any three consecutive t

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$ac + bd > b^2 + c^2$

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(C) If $a, b, c, d$ are in $H.P.$,then their reciprocals $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \frac{1}{d}$ are in $A.P.$For any three consecutive terms in $H.P.$,the middle term is the harmonic mean of the other two. Thus,$b = \frac{2ac}{a+c}$ and $c = \frac{2bd}{b+d}$.We know that for any two positive numbers,the Geometric Mean $(G.M.)$ is greater than the Harmonic Mean $(H.M.)$.For $a$ and $c$,the $G.M.$ is $\sqrt{ac}$ and the $H.M.$ is $b$. Therefore,$\sqrt{ac} > b \Rightarrow ac > b^2$.Similarly,for $b$ and $d$,the $G.M.$ is $\sqrt{bd}$ and the $H.M.$ is $c$. Therefore,$\sqrt{bd} > c \Rightarrow bd > c^2$.Adding these two inequalities,we get $ac + bd > b^2 + c^2$.

If $a, b, c, d$ are in $H.P.$,then

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