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If $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are three positive numbers in an arithmetic progression, then:
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Verified Answer
The correct answer is:
$\mathrm{ab}+\mathrm{bc} \geq 2 \mathrm{ac}$
Since, $\mathrm{a}, \mathrm{b}, \mathrm{c}$, are in $\mathrm{AP}$
$\frac{1}{\mathrm{a}}, \frac{1}{\mathrm{~b}}, \frac{1}{\mathrm{c}}$ are in $\mathrm{HP}$.
Since, $\mathrm{AM} \geq \mathrm{HM}$
$\Rightarrow \quad \mathrm{b} \geq \frac{2 \mathrm{ac}}{\mathrm{a}+\mathrm{c}}$
[since $\mathrm{AM}=\mathrm{b}$ and $\left.\mathrm{HM}=\frac{2 \mathrm{ac}}{\mathrm{a}+\mathrm{c}}\right]$
$\Rightarrow \quad \mathrm{ab}+\mathrm{bc} \geq 2 \mathrm{ac}$
$\frac{1}{\mathrm{a}}, \frac{1}{\mathrm{~b}}, \frac{1}{\mathrm{c}}$ are in $\mathrm{HP}$.
Since, $\mathrm{AM} \geq \mathrm{HM}$
$\Rightarrow \quad \mathrm{b} \geq \frac{2 \mathrm{ac}}{\mathrm{a}+\mathrm{c}}$
[since $\mathrm{AM}=\mathrm{b}$ and $\left.\mathrm{HM}=\frac{2 \mathrm{ac}}{\mathrm{a}+\mathrm{c}}\right]$
$\Rightarrow \quad \mathrm{ab}+\mathrm{bc} \geq 2 \mathrm{ac}$
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