1A.M of nth powers > nth power of A.M
\left(\frac{a^{n}+c^{n}}{2} \right)>\left(\frac{a+c}{2} \right)^{n}\Rightarrow a^{n}+c^{n}>2\left(\frac{a+c}{2} \right)^{n} ....................(1)
but we know AM >GM>HM
so AM > HM
a+b2 > b
= \left( \frac{a+c}{2}\right)^{n}>b^{n}
and
= 2\left( \frac{a+c}{2}\right)^{n}>2b^{n} ......................(2)
from (1) and (2)
we get
a^{n}+c^{n}>2b^{n} [ Proved ]
