Question

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

Answer 1

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