Last Updated : 19 Apr, 2024

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** AM-GM Inequality **is one of the most famous inequalities in algebra. Before going through AM-GM Inequality first we need to go through arithmetic mean and geometric mean concepts.

Arithmetic mean is defined as the sum of all the quantities divided by the number of quantities.**Arithmetic Mean:**Geometric mean is defined as the mean which is calculated by multiplying the n number together and then taking their (1/n)**Geometric Mean:**^{th }root.

In this article, we will learn about **AM-GM Inequality, the relationship between AM and GM, and solve examples and problems on it.**

## What is AM-GM Inequality?

The Arithmetic Mean-Geometric Mean (AM-GM) Inequality is a fundamental result in algebra that provides a relationship between the arithmetic mean and the geometric mean of a set of non-negative real numbers. This inequality states that for any list of non-negative real numbers, the arithmetic mean (average) is at least as great as the geometric mean.

**Read More: **

- Arithmetic Mean
- Geometric Mean

## AM–GM Inequality Relationship

AM–GM Inequality is discussed below in the article,

For two positive numbers a and b

Arithmetic Mean: A.M = (a+b)/2

Geometric Mean: G.M = √(ab)

A.M ≥ G.M

(a+b)/2 ≥ √(ab)

For ‘n’ positive numbers a_{1}, a_{2}, a_{3}, a_{4}, … a_{n}

Arithmetic Mean: A.M = (a

_{1}+ a_{2}+ a_{3}+ a_{4}+………. +a_{n}) / nGeometric Mean: G.M = (a

_{1 }a_{2}a_{3 }a_{4}………. a_{n})^{1/n}A.M ≥ G.M

(a_{1}+ a_{2}+ a_{3}+ a_{4}+………. +a_{n}) / n ≥ (a_{1 }a_{2}a_{3 }a_{4}………. a_{n})^{1/n}

## AM-GM Inequality Formula

For two positive numbers a and b,

(a+b)/2 ≥ √(ab)See AlsoFiltru Combustibil Original Mercedes Benz CLA, A, B, G, M, GLA - eMAG.roপত্রিকা (২০শে এপ্রিল): 'সারা দেশে হিট অ্যালার্ট জারি' - BBC News বাংলাRelationship between A.M. and G.M.- Study Material for IIT JEEWhat does A-B:AO and G-M: GO mean on A7R3 LCD screen: Sony Alpha Full Frame E-mount Talk Forum: Digital Photography ReviewFor ‘n’ positive numbers a

_{1}, a_{2}, a_{3}, a_{4}, … a_{n}

(a1+ a2+ a3+ a4+………. +an) / n ≥ (a1 a2 a3 a4 ………. an) 1/n

## AM–GM Inequality Relationship Proof

Statement: For any n positive numbers a_{1}, a_{2}, … a_{n} Arithmetic Mean is always greater than equal to Geometric Mean. **A.M ≥ G.M**

**Proof:**

For two numbers,

A.M – G.M = (a+b)/2 – √(ab)

A.M – G.M = ½ (a+b – 2 √(ab) )

A.M – G.M = ½ (√a – √b)

^{2}We know that square of any number is positive,( i.e ≥0) Hence,

A.M – G.M ≥0

A.M ≥ G.M

Hence we conclude by above proof that for all positive Numbers, A.M ≥ G.M

**People Also View:**

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- Arithmetic Progression
- Geometric Progression

## Solved Example on AM-GM Inequality

**Example 1: Find the arithmetic mean of 3 and 27**

**Solution:**

Arithmetic Mean: A.M = (a+b)/2

A.M = (3+27)/2

A.M = 15

**Example 2: Find the Geometric Mean of 3 and 27**

**Solution:**

Geometric Mean: G.M = √(ab)

G.M = √(27 × 3) = √81

G.M = 9

**Example 3: If x>0, then Prove That: x+ (1/x) ≥ 2**

**Solution:**

Since x>0 we can apply A.M-G.M Inequality here,

A.M ≥ G.M

(x+1/x) / 2 ≥ (x . 1/x)

^{½}(x+1/x) /2 ≥ 1

x+(1/x) ≥ 2 (Proved)

**Example 4: If x,y>0,then Prove That: x**^{2}**+y**^{2}** ≥ 2xy**

**Solution:**

Since x,y>0 we can apply A.M-G.M Inequality here,

A.M ≥ G.M

For two variables x and y

(x+y) /2 ≥ (xy)

^{1/2}squaring both sides-

(x+y)

^{2}/4 ≥ xyx

^{2}+y^{2}≥ 2xy

**Example 5: If x,y,z ≤ 0,then can we Prove That: (x+y)(y+z)(z+x) ≥ 8xyz through A.M-G.M Inequality**

**Solution:**

Since x,y,z ≤ 0

So, we can’t apply A.M-G.M Inequality here,

**Example 6: If a,b,c ∈ R+, such that a+b+c = 3, find the maximum value of abc.**

**Solution:**

Since a,b,c >0 we can apply A.M-G.M Inequality here.

We need to find the value of the product of a,b,c i.e abc.

Applying A.M ≥ G.M,

(a+b+c)/3 ≥

^{3}√(abc)3/3 ≥ (abc)

^{1/3}1 ≥ (abc)

^{1/3}cubing both sides, 1

^{3}≥ (abc)so, abc ≤1

Hence the maximum value of abc = 1

## Summary – AM-GM Inequality

The AM-GM Inequality is a fundamental mathematical principle stating that for any set of non-negative real numbers, the arithmetic mean (AM) is always greater than or equal to the geometric mean (GM). Specifically, the inequality formula is expressed as **(a**_{1}**+ a**_{2}**+ a**_{3}**+ a**_{4}**+………. +a**_{n}**) / n ≥ (a**_{1 }**a**_{2}** a**_{3 }**a**_{4}** ………. a**_{n}**) **** ^{1/n}** where

**a**

_{1}

**, a**

_{2}

**, …….a****are non-negative numbers. The equality holds if and only if all the numbers in the set are equal. This inequality is crucial in various mathematical contexts, especially in proving bounds and optimizing algebraic expressions. It finds applications across diverse fields such as economics, engineering, and optimization problems, making it a versatile and powerful tool in theoretical and applied mathematics.**

_{n}## FAQs on AM-GM Inequality

**What is AM-GM inequality?**

**What is AM-GM inequality?**

AM -GM inequality states that for any n positive numbers a

_{1}, a_{2}, … a_{n}Arithmetic Mean is always greater than equal to Geometric Mean. i.e.

AM ≥ GM

**Can we apply this concept if signs of some numbers are unknown or Non-Positive?**

**Can we apply this concept if signs of some numbers are unknown or Non-Positive?**

No, since we aren’t sure of the numbers to be positive we cant apply this concept.

**What is Arithmetic Mean?**

**What is Arithmetic Mean?**

Arithmetic mean is defined as the sum of all the quantities divided by the number of quantities. It is also called the average.

**What is Geometric Mean?**

**What is Geometric Mean?**

Geometric mean is defined as the mean which is calculated by multiplying the n number together and then taking their (1/n)

^{th }root.

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