Linked Questions

7
votes
2answers
3k views

Proving the AM:GM inequality [duplicate]

I am doing past exam papers preparing for the finals and I came across this questions about three times: Prove that: $$\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\geq \sqrt[n]{a_{1}.a_{2}...a_{n}}$$ ...
-2
votes
4answers
495 views

A.M.>G.M. of four numbers [duplicate]

Prove that arithmetic mean of $4$ numbers is greater than geometric mean of the same $4$ numbers, i.e. prove that $$\dfrac{a+b+c+d}{4} > (abcd)^{\frac1{4}}$$
1
vote
3answers
605 views

Proof of AM-GM inequality for $n=3$: $\frac{a+b+c}{3}\geq\sqrt[3]{abc}$ [duplicate]

Sorry for bad formatting, I couldn't mark the 3rd root on the right hand side... I've figured this out into the point where (and yeah, the problem is to prove that this applies to all non-negative ...
3
votes
2answers
119 views

$ x \ge 0\text{ and } y \ge 0 \implies \frac{x+y}{2} \ge \sqrt{xy} $ [duplicate]

The above applies $\forall x,y \in \mathbb{R}$ I've tried: $x + y \ge 0$ $$x + y \ge x$$ $$ (x + y)^2 \ge 2xy$$ $$\frac{(x + y)^2}{2} \ge xy$$ But the closest I get is $\dfrac{x+y}{\sqrt{2}} \ge \...
2
votes
1answer
309 views

A question on mean value inequality [duplicate]

It is known that mean value inequality is very useful. It is: For any $0 \le a_i (i=1,2,\dots,n)$, $$ a_1 a_2\dots a_n\le (\frac{a_1+a_2+\dots + a_n}{n})^n \tag1 $$ My question is: how many ...
1
vote
3answers
170 views

Arithmetic Mean And Geometric Mean [duplicate]

If $A.M$ and $G.M$ are Arithmetic Mean And Geometric Mean respectively then prove that $A.M \ge G.M$. My Attempt : Let $a$ and $b$ are any two real positive numbers. Then: $$A.M=\frac{a+b}{2}$$ $$...
-1
votes
1answer
192 views

Inequality of arithmetic and geometric means [duplicate]

Prove that if $x_1,x_2,...,x_n$ are positive numbers, then $$\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...\frac{1}{x_n}}\leq \sqrt[n]{x_1\cdot x_2\cdot ... \cdot x_n}\leq\frac{x_1+x_2+...+x_n}{n}$$
1
vote
1answer
168 views

How to prove by Mathematical Induction. [duplicate]

I want to know how to prove this inequality by mathematical induction: $a_k's$ are nonnegative numbers. Prove that$$a_1a_2\cdots a_n\leq \left(\frac{a_1+a_2+\cdots+a_n}{n}\right)^n.$$ In the inductive ...
-2
votes
1answer
153 views

Prove by induction $\frac{a_{1} + a_{2} + a_{3} +…+ a_{n}}{n} \geq \sqrt[n]{a_1 \cdot a_2 \cdot a_3 \cdot …a_n}$. [duplicate]

Let $a_1$ $a_2$,..., $a_n$ be positive numbers. Prove that $\frac{a_{1} + a_{2} + a_{3} +...+ a_{n}}{n} \geq \sqrt[n]{a_1 \cdot a_2 \cdot a_3 \cdot ....a_n}$. Mine is about trying to understand how ...
2
votes
2answers
124 views

Proving the AM-GM Inequality? [duplicate]

Could anyone provide a simple, easily digestible, and well-explained proof of the AM-GM inequality (the multi-value version)? Much appreciated.
1
vote
1answer
87 views

If AM-GM holds for any $2^n$ positive real numbers, show that AM-GM holds for *any number* of positive real numbers. [duplicate]

I was able to prove by induction on $n\in\mathbb{N}$ that for any $m:=2^n$ positive real numbers we have $\text{Geometric Average}\leq\text{Arithmetic Average}$, I.e. $\sqrt[m]{\Pi_{i=1}^m a_i} = $$\...
0
votes
0answers
70 views

Prove that $ a_1 + a_2 + a_3 + \cdots + a_n \geq n$ if $a_1 a_2 \cdots a_n = 1$ [duplicate]

Prove that for any real numbers $a_1, a_2,\ldots, a_n > 0$ $(n \in \mathbb{N}, n \geq 2)$ satisfying $a_1 a_2 \cdots a_n = 1$ we have $ a_1 + a_2 + a_3 + \cdots + a_n \geq n$.
0
votes
0answers
21 views

Proof for inequality containing a summatory and a product [duplicate]

Given the following inequality: $$\sqrt[N]{\prod_i^N x_i} < \frac{1}{N} \sum_i^N x_i$$ for $x_i \in \mathbb{R^+_0}$ (positive reals) and $N \in \mathbb{N}^+$ (positive integer). How can I prove ...
9
votes
7answers
4k views

What is the theorem that has the most proofs?

Classical theorems like the irrationality of $\sqrt{2}$ or the infinitude of the primes have lots of proofs. But one theorem in particular, which I studied years ago in an introductory course of ...
10
votes
5answers
3k views

Geometric mean never exceeds arithmetic mean

This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.) The geometric ...

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