# Thread: Product of n numbers = 1, sum >= n - Proof

1. ## Product of n numbers = 1, sum >= n - Proof

Hi everyone,

I have a little problem with the following exercise.

I have to prove by mathematical induction that for all n real numbers, which’s product is 1, their sum is greater than or equal to n.

I’d be very happy if someone could help me with this.

2. Originally Posted by thomasdotnet
Hi everyone,

I have a little problem with the following exercise.

I have to prove by mathematical induction that for all n real numbers, which’s product is 1, their sum is greater than or equal to n.

I’d be very happy if someone could help me with this.
Let n be 2. We'll use -2 and -0.5 as our real numbers. The product is 1. Their sum is -2.5 which is not greater than or equal to n.

Are you sure you copied the problem correctly? Does it say positive real numbers?

3. Hello thomasdotnet.

Welcome to Math Help Forum!
Originally Posted by thomasdotnet
Hi everyone,

I have a little problem with the following exercise.

I have to prove by mathematical induction that for all n real numbers, which’s product is 1, their sum is greater than or equal to n.

I’d be very happy if someone could help me with this.
Assuming that the numbers are all non-negative, the easiest proof is to use the AM-GM inequality: that the arithmetic mean of a set of non-negative real numbers is greater than or equal to their geometric mean.

If the product of the $\displaystyle n$ numbers is $\displaystyle 1$, then their GM is $\displaystyle 1$. Hence their $\displaystyle AM \ge 1 \Rightarrow$ their sum $\displaystyle \ge n$.

The proof by Induction of this inequality can be found here: Inequality of arithmetic and geometric means - Wikipedia, the free encyclopedia

Grandad

4. Actually I did copy the question correctly, but what you said wytiaz makes sense, so think that it's only about positive real numbers. Thanks.

Thank you for pointing me to the AM-GM inequality Grandad, that's exactly what I needed.