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.
Hello thomasdotnet.
Welcome to Math Help Forum!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 numbers is , then their GM is . Hence their their sum .
The proof by Induction of this inequality can be found here: Inequality of arithmetic and geometric means - Wikipedia, the free encyclopedia
Grandad