A) \[{{a}^{n}}+{{c}^{n}}<2{{b}^{n}}\]
B) \[{{a}^{n}}+{{c}^{n}}>2{{b}^{n}}\]
C) \[{{a}^{n}}+{{c}^{n}}=2{{b}^{n}}\]
D) None of the above
Correct Answer: B
Solution :
For two numbers \[a\] and \[c\] \[\frac{{{a}^{n}}+{{c}^{n}}}{2}>{{\left( \frac{a+c}{2} \right)}^{n}}\] (Where\[n\in N,\ n>1\]) \[\because \]\[A.M.>G.M.>H.M.\] \[\therefore \]\[\frac{a+b}{2}>b\] \[(\because \ a,\ b,\ c\] are in H.P.) \[\Rightarrow \]\[{{\left( \frac{a+c}{2} \right)}^{n}}>{{b}^{n}}\]\[\Rightarrow \]\[\frac{{{a}^{n}}+{{c}^{n}}}{2}>{{\left( \frac{a+c}{2} \right)}^{n}}>{{b}^{n}}\].
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