Question

If A, G and H are A.M, G.M and H.M. of any two given positive numbers, then find the relation between A, G and \(\mathrm{H}\). (1) \(\mathrm{A}^{2}=\mathrm{GH}\) (2) \(\mathrm{G}^{2}=\mathrm{AH}\) (3) \(\mathrm{H}^{2}=\mathrm{AG}\) (4) \(\mathrm{G}^{3}=\mathrm{A}^{2} \mathrm{H}\)

Short answer

Answer: A² = GH

Step-by-step solution

Write expressions for A, G, and H

Let the two given positive numbers be x and y. Then, A (Arithmetic Mean) = \(\frac{x + y}{2}\) G (Geometric Mean) = \(\sqrt{xy}\) H (Harmonic Mean) = \(\frac{2xy}{x + y}\)

Eliminate x and y from the expressions

To find a relation between A, G, and H, we need to eliminate x and y from the above expressions. We can square the expressions of A and H, which helps eliminate the denominators. A² = \(\frac{(x+y)²}{2^2}\) = \(\frac{x^2 + 2xy + y^2}{4}\) H² = \(\frac{(2xy)^2}{(x+y)^2}\) = \(\frac{4x^2y^2}{(x^2 + 2xy + y^2)}\) Now, we can manipulate these equations to find the correct relationship.

Test each given option

(1) Test A² = GH A² = G² \(\frac{x^2 + 2xy + y^2}{4}\) = \(xy\) \(x^2 + 2xy + y^2 = 4xy\) \(x^2 - 2xy + y^2 = 0\) \((x - y)^2 = 0\) Since x and y are positive numbers, this relation is true, so this is our answer. (2) Test G² = AH Since we found the correct answer in option 1, we don't need to test the remaining options. (3) Test H² = AG Since we found the correct answer in option 1, we don't need to test the remaining options. (4) Test G³ = A²H Since we found the correct answer in option 1, we don't need to test the remaining options. So the correct relationship between A, G, and H is: A² = GH. That's option (1).

Key concepts

A.M., G.M., H.M. Relation

Understanding the relationship between the Arithmetic Mean (A.M.), Geometric Mean (G.M.), and Harmonic Mean (H.M.) is crucial when working with sequences and series of numbers. Consider two positive numbers, for which A.M. is their average, G.M. is the square root of their product, and H.M. is the reciprocal of the average of their reciprocals. That gives us the foundational formulas:

• A (Arithmetic Mean) = \frac{x + y}{2}
• G (Geometric Mean) = \(\sqrt{xy}\)
• H (Harmonic Mean) = \(\frac{2xy}{x + y}\)

To discern the relationship amongst A, G, and H, we can manipulate the equations. The key is to eliminate the individual numbers (x and y) by focusing on how A, G, and H are interrelated. Square the equations of A and H to facilitate the elimination process:

• A² = \(\frac{(x+y)²}{4}\)
• H² = \(\frac{(2xy)²}{(x+y)²}\)

After careful manipulation, we discover the correct relationship is A² = GH. This equation represents a fundamental property of means, which states that the square of the arithmetic mean of two numbers is equal to the product of their geometric and harmonic means, illustrating a beautiful symmetry in mathematics.

Quadratic Equations

Quadratic equations play a pivotal role in various areas of mathematics and applications. An equation of the form \( ax^2 + bx + c = 0 \) is termed as a quadratic equation, where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). These equations are integral in finding relationships among means, as demonstrated in the relationship between A.M., G.M., and H.M.

The solutions to a quadratic equation are found using various methods, including factorization, completing the square, or the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

These solutions are also known as roots, which are the x-values where the parabola represented by the quadratic equation intersects with the x-axis. Understanding quadratic equations is not only crucial for this particular concept of mean relationships but also for solving a wide range of problems in algebra, geometry, physics, and engineering.

Square and Cube Roots

Square and cube roots are essential mathematical operations that are closely related to exponents. A square root of a number 'a' is a number 'x' that, when multiplied by itself, gives 'a': \( x^2 = a \). Conversely, a cube root of 'a' is a number 'x' that, when used three times in a multiplication (cubed), equals 'a': \( x^3 = a \).

For the geometric mean (G.M.), the square root function is used in its calculation, as it represents the square root of the product of two numbers. The use of square and cube roots is pervasive in algebra and is crucial for solving equations that involve powers higher than one, which includes quadratic equations requiring square roots for their solutions. Similarly, understanding cube roots is necessary when dealing with cubic equations and geometry involving three-dimensional shapes.