If a , b , c are in H . P . show that
$a^{n} + a^{c} > 2 b^{n}, n$ begin a + live integer

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Solution

We know that $\frac{a^{n} + c^{n}}{2} {(\frac{a + c}{2})}^{2}$ Now since a , b , c are in H . P . is the H . M between a and c . Also $\frac{a + c}{2}$ is the A . M between a and c and A > H . Hence $\frac{a + c}{2} a n d s o {(\frac{a + c}{2})}^{n} > b^{n}$
From (1) and (2) we get
$\frac{a^{n} + c^{n}}{2}$ > $b^{n} i . e . a^{n}$ $+ c^{n} > 2 b^{n}$

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