AM GM Inequality

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: Arithmetic mean is defined as the sum of all the quantities divided by the number of quantities.
• 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.

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 a₁, a₂, a₃, a₄, … a_n

Arithmetic Mean: A.M = (a₁+ a₂+ a₃+ a₄+………. +a_n) / n

Geometric Mean: G.M = (a₁ a₂ a₃ a₄ ………. a_n) ^1/n

A.M ≥ G.M

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

AM-GM Inequality Formula

For two positive numbers a and b,

(a+b)/2 ≥ √(ab)

For ‘n’ positive numbers a₁, a₂, a₃, a₄, … 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 a₁, a₂, … 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:

• 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: x²+y² ≥ 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 ≥ xy

x²+y²  ≥ 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, 1³ ≥ (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 (a₁+ a₂+ a₃+ a₄+………. +a_n) / n ≥ (a₁ a₂ a₃ a₄ ………. a_n) ^1/n where a₁, a₂, …….a_n 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.

FAQs on  AM-GM Inequality

What is AM-GM inequality?

AM -GM inequality states that for any n positive numbers a₁, a₂, … 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?

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

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?

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