Question 1

Let A = $\displaystyle {x_1, x_2, ......x_n} $ be a set of positive numbers in geometric progression. Suppose n is odd. Show that the geometric mean and the median of A are the same.

Isn't the geometric mean = n th root of $\displaystyle \frac{a(1-r^n)}{1-r} $ ? And the median is $\displaystyle \frac{1}{2} ( x_1 + x_n) $ ?

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Question 2

Let A = $\displaystyle {x_1, x_2, ......x_n} $ be a set of positive numbers with arithmetic mean, geometric mean and harmonic mean A, G, H respectively.

In the case n=2, prove that A ≥ G ≥ H.

Help will be much appreciated. Thank you.