# Arithmetic Mean-Geometric Mean Inequality

This post is available in PDF. This topic is solely in problem solving in am-gm inequality. The proof for this inequality can be found here.

Thanks to Engr. Roy Roque Rivera for this very useful article.

** Problem 1**: Show that if , then

Solution: By AM – GM Inequality, we have

Hence,

or simply

** Problem 2** (AM – HM Inequality): Show that if , then

**Solution**: By AM – GM Inequality, we have

which can be written as

But from AM – GM Inequality, we know that that geometric mean is less than or equal to the arithmetic mean; hence, we obtain

Comment: In general, for positive real numbers , the following inequality holds

**Problem 3**: Show that if , then

**Solution 1**:By AM – GM Inequality, we have

Hence,

Simplifying gives

Thus,

**Solution 2**: By AM – HM Inequality, we know that

Which can be easily written into

**Problem 4:** Show that if , then

**Solution 1:** By AM – GM Inequality, we have

*

**

***

Adding *,**, and ***, gives us

Hence,

**Solution 2**: By AM – HM Inequality, we know that

Which can be easily written into

**Problem 5:** Show that if , then

**Solution:** By AM – GM Inequality, we have

*

**

***

Adding *, **, *** gives us

Hence,

More problems to page 2.

### Dan

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