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Thu 8 Sep, 2005 07:49 pm
The arithmetic mean and the geometric mean of the positive real numbers a_1,a_2, . . . , a_n are A = (a_1 + a_2 + . . . + a_n) / n and
G = (a_1 + a_2 + . . . + a_n) ^ (1/n), respectively. Use mathematical induction to prove that A >= G for every finite sequence of positive real numbers. When does equality hold?
Quick check: The geometric mean is typically:
(a1*a2*a3*...an)^(1/n)
Is that what you intended?
Well, I have spent way to much time on this one and can't find the path. I can prove (via induction) that equality holds when all elements are equal to the same value.
I'm stuck at proving that (n+y)/(n+1) >= y^(1/1+n) for positive values of y.
If someone can figure that out, I have the rest.
That's exactly where I'm stuck...
engineer wrote:Quick check: The geometric mean is typically:
(a1*a2*a3*...an)^(1/n)
Is that what you intended?
Yes, you're right. I did a typo. Thanks!
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