a, b, c are in H.P. and n does not lie between 0 and 1 then $a^{n} + c^{n} > 2 b^{n}$

True

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False

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Solution

True
$\frac{a^{n} + c^{n}}{2} > {(\frac{a + c}{2})}^{n}$
If n does not lie between 0 and 1
But we know that A.M. > G.M. > H.M.
$∴ \frac{a + c}{2}$ the A.M. of a and c is >b i.e. H.M. of a and c because a, b, c are given to be in H.P.
$∴ {(\frac{a + c}{2})}^{n} > b^{n}$
$∴ \frac{a^{n} + c^{n}}{2} > {(\frac{a + c}{2})}^{n} > b^{n}$
$∴ a^{n} + c^{n} > 2 b^{n}$

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