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If $a, b, c$ are in H.P., then for all $n \in N$ the true statement is
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Verified Answer
The correct answer is:
$a^n+c^n \gt 2 b^n$
For two numbers $a$ and $c$
$\frac{a^n+c^n}{2} \gt \left(\frac{a+c}{2}\right)^n$ (Where $n \in N, n \gt 1$)
$A.M. \gt G.M. \gt H.M.$
$\therefore \quad \frac{a+b}{2} \gt b$ $a, b, c$ are in H.P.)
$\Rightarrow\left(\frac{a+c}{2}\right)^n \gt b^n \Rightarrow$ $\frac{a^n+c^n}{2} \gt \left(\frac{a+c}{2}\right)^n \gt b^n$
$\frac{a^n+c^n}{2} \gt \left(\frac{a+c}{2}\right)^n$ (Where $n \in N, n \gt 1$)
$A.M. \gt G.M. \gt H.M.$
$\therefore \quad \frac{a+b}{2} \gt b$ $a, b, c$ are in H.P.)
$\Rightarrow\left(\frac{a+c}{2}\right)^n \gt b^n \Rightarrow$ $\frac{a^n+c^n}{2} \gt \left(\frac{a+c}{2}\right)^n \gt b^n$
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