# Questions tagged [means]

In probability and statistics, mean and expected value are used synonymously to refer to one measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. For a data set, refers to a central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values.

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### Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual ...
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### Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$

I am having trouble with this problem from my latest homework. Prove the arithmetic-geometric mean inequality. That is, for two positive real numbers $x,y$, we have $$\sqrt{xy}≤ \frac{x+y}{2} .$$ ...
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### Providing that: $\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$

Let $a$ and $b$ be positive reals. Show that $$\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$$
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### Prove QM-AM inequality

$$\dfrac{x_1^2+ x_2^2 + \cdots + x_n^2}n \geq \left(\dfrac{x_1+x_2+\cdots+x_n}n\right)^2$$ I don't think AM, GM can be used here. And simple expansion doesn't help too. What should I do?
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### How to show $\lim_{n \to \infty} \sqrt[n]{a^n+b^n}=\max \{a,b\}$? [duplicate]

Let $a\geq 0$ and $b\geq 0$. Prove that $\lim_{n \to \infty} \sqrt[n]{a^n+b^n}=\max \{a,b\}$. [Hint: Use the identity $(a^n -b^n)=(a-b)(\sum_{i=0}^{n-1}a^ib^{n-1-i})$] I need some help! I cannot ...
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### Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$

I am searching for an elementary proof of the AM-GM inequality in three variables: $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$ The inequality of the geometric mean vs the arithmetic mean of two variables ...