SOLUTION: prove that the geometric mean of two positive, unequal numbers is less than their arithmetic mean.
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Question 982143: prove that the geometric mean of two positive, unequal numbers is less than their arithmetic mean.
Answer by ikleyn(52790) (Show Source): You can put this solution on YOUR website!
Let a and b be two positive, unequal real numbers.
Then their arithmetic mean is and geometric mean is .
We need to prove that
>= .
Let us start with this inequality
> 0,
which is always true for any real unequal numbers a and b, because the left part is the square of the real number , which is unequal to zero.
Rewrite this inequality in this way step by step:
> , (apply the formula of the square of a difference)
> , (move the term from the left side to the right with the opposite sign)
> .
The last inequality is exactly what you have to prove.
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