Show that for positive real numbers such that , then

.

Consider the average of the positive numbers *x* and *y*: . The AM-GM inequality tells us that

and so for any positive numbers *x* and *y*. Letting and , we then see that for all *i*=1,2,…,*n*. Multiplying these *n* inequalities together:

,

where the last is because .

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Tags: AM-GM Inequality, Math, Monday Math

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