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Question

Let a1,a2...an be a set of n positive real number (not all equal),
Let mbe a real number then the arithmetic mean of mth power is counted as
(i):am1+am2+am3+...+am1n>(a1+a2+...+ann)m if the real number mR[0,1]
(ii): am1+am2+am3+...+am1n<(a1+a2+...+ann)m if m(0,1) and equality holds if m=0 or m=1.
On the basis of above information answer the following questions,If sum of the mth powers of first n natural numbers λm>1 then

A
λ2m>(n1/m.n+1)m
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B
2m.λ>(n1/m.(n+1))m
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C
2mλ>(n+(n+1))m
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D
None of these
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Solution

The correct option is B 2m.λ>(n1/m.(n+1))m
For m>1
A.M. of mth powers >mth powers of A.M
1m+2m+...+nmn>(1+2+...+nn)m
λn>(n+12)m
2m.λ>(n1/m.(n+1))m
Ans: B

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