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## Homework Statement

The problem is stated as follows:

"The result in Problem 1-7 has an important generalization: If ##a_1,...,a_n≥0##, then the "arithmetic mean" ##A_n=\frac {a_1+...+a_n} {n}##

and "geometric mean"

##G_n=\sqrt[n] {a_1...a_n}##

Satisfy

##G_n≤A_n##

Suppose that ##a_1\lt A_n##. Then some ##a_i## satisfies ##a_i\gt A_n##; for convenience, say ##a_2\gt A_n##. Let ##\bar a_1=A_n## and let ##\bar a_2=a_1+a_2-\bar a_1##. Show that ##\bar a_1 \bar a_2≥a_1a_2##.

Why does repeating this process enough times eventually prove that ##G_n≤A_n##? (This is another place where it is a good exercise to provide a formal proof by induction, as well as an informal reason.) When does equality hold in the formula ##G_n≤A_n##?"

2. Homework Equations

2. Homework Equations

None that I can think of, perhaps except ##\sqrt {ab} ≤ \frac {a+b} 2## since the AM-GM inequality is an extension of this but I doubt I will actually use it here.

## The Attempt at a Solution

I've proved ##\bar a_1 \bar a_2≥a_1a_2## fairly easily, my problem is with what comes after that. What does it mean "repeating the process"? What do I do for the other ##a_i##? I don't know how the "bar" is defined here...

If I replace ##a_1,a_2## with ##\bar a_1,\bar a_2## I can see that the arithmetic mean doesn't change and the geometric mean grows or doesn't change, but that's all I can really deduce since I don't understand the question, so I would love so assistance.

Thanks in advance to all the helpers.