If $a, b, c$ are in H.P. and they are distinct and positive, then prove that $a^{n} + c^{n} > 2 b^{n}$

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Solution

Let $a^{n}$ and $c^{n}$ be two numbers
then $\frac{a^{n} + c^{n}}{2} > {(a^{n} c^{n})}^{1 / 2}$
$a^{n} + c^{n} > 2 {(a c)}^{n / 2}$ (i)
Also G.M.> H.M.
i.e. $\sqrt{a c} > b, {(a c)}^{n / 2} > b^{n}$ (ii)
hence from (i) and (ii), we ge $a^{n} + c^{n} > 2 b^{n}$

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