How to prove by induction that the if the sum of n positive numbers is n than their product is smaller equal to 1?

Question

Hi what is a way to prove that if we have n positive numbers which sum is n than their product is smaller equal to 1?

Answer 1

The question is more clearly phrased as follows:

Given $n$ positive real numbers $a_1,a_2, \ldots, a_n$ such that $a_1+a_2+\ldots+a_n=n$, prove that:

$$a_1a_2\cdot \ldots \cdot a_n\leq 1$$

This is pretty much a direct application of the AM-GM inequality for $n$ numbers, that is:

$$\frac{a_1+a_2+\ldots +a_n}{n} \geq \sqrt[n]{a_1a_2\cdot \ldots \cdot a_n}$$

With the given condition, this implies:

$$1 \geq \sqrt[n]{a_1a_2\cdot \ldots \cdot a_n}$$

and now raise to the $n$th power to get the desired conclusion.