prove the following inequation is alway true?

Question

(x[1]+x[2]+x[3]+....+x[n-1]+x[n])^(1/(n)) < (x[1]+x[2]+x[3]+....+x[n-1]+x[n])/(n)

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http://img100.imageshack.us/img100/2145/image006dv8.jpg

Answer 1

he might want us to prove for x is integer numbers x>0

Answer 2

It isn't always true. For example, let n=1 and x=0.36. In fact, we must have x[1] > 1 in order for the n=1 case of the inequality to be true.

And x[1] > 1, x[2] > 1 is not enough for n=2. Let x[1]=x[2]=1.5; then

sqrt(x[1]+x[2]) = sqrt(3) ≈ 1.732, but (x[1] +x[2])/2 = 1.5

There must be some additional restrictions on the x's.

Answer 3

actually this is not always true
we can prove it as follows
(segma(x[n]) )^ 1/n < segma (x[n]) /n
so make both sides to the power of n
so
segma(x[n]) < segma (x[n])^n /n^n
n^n < segma(x[n] )^(n-1)
take log of both sides
n log n < (n-1) log segma(x[n])
n/(n-1) < log(segma x[n]) /log (n)
as n/(n-1) >1 then log (segma x[n])/log n must be >1
that means
log segma x[n] > log n
segma x[n] > n
so the restriction we have is that segma x[n] must be > n