algebra

Relationship Between Arithmetic Mean, Harmonic Mean, and Geometric Mean of Two Numbers

For two numbers [math]x[/math] and [math]y[/math], let [math]\,x, \, a, \, y\,[/math] be a sequence of three numbers. If [math]\,x, \, a, \, y\,[/math] is an arithmetic progression then [math]a[/math] is called arithmetic mean. If [math]\,x, \, a, \, y\,[/math] is a geometric progression then [math]a[/math] is called geometric mean. If [math]\,x, \, a, \, y\,[/math] form a harmonic progression then [math]a[/math] is called harmonic mean.

Let [math]AM[/math] = arithmetic mean, [math]GM[/math] = geometric mean, and [math]HM[/math] = harmonic mean. The relationship between the three is given by the formula

AM \times HM = GM^2

Below is the derivation of this relationship.

Derivation of Sum of Finite and Infinite Geometric Progression

Geometric Progression, GP

Geometric progression (also known as geometric sequence) is a sequence of numbers where the ratio of any two adjacent terms is constant. The constant ratio is called the common ratio, [math]r[/math] of geometric progression. Each term therefore in geometric progression is found by multiplying the previous one by [math]r[/math].

Eaxamples of GP:

  • [math]3, \, 6, \, 12, \, 24, \dots \,[/math] is a geometric progression with [math]r = 2[/math]
  • [math]10, \, -5, \, 2.5, \, -1.25, \dots \,[/math] is a geometric progression with [math]r = -\frac{1}{2}[/math]

Derivation of Sum of Arithmetic Progression

Arithmetic Progression, AP

Definition

Arithmetic Progression (also called arithmetic sequence), is a sequence of numbers such that the difference between any two consecutive terms is constant. Each term therefore in an arithmetic progression will increase or decrease at a constant value called the common difference, d.

Examples of arithmetic progression are:

  • 2, 5, 8, 11,... common difference = 3
  • 23, 19, 15, 11,... common difference = -4

Sum and Product of Roots

The quadratic formula

[display]x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}[/display]

give the roots of a quadratic equation which may be real or imaginary. The [math]\pm[/math] sign in the radical indicates that

[math]x_1 = \dfrac{-b + \sqrt{b^2-4ac}}{2a}[/math] and [math]x_2 = \dfrac{-b - \sqrt{b^2-4ac}}{2a}[/math]

where [math]x_1[/math] and [math]x_2[/math] are the roots of the quadratic equation [math]ax^2 + bx + c = 0[/math]. The sum of roots [math]x_1 + x_2[/math] and the product of roots [math]x_1 \, x_2[/math] are common to problems involving quadratic equation.

Derivation of Quadratic Formula

The roots of a quadratic equation [math]ax^2 + bx + c = 0[/math] is given by the quadratic formula

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The derivation of this formula can be outlined as follows:

  1. Divide both sides of the equation [math]ax^2 + bx + c = 0[/math] by [math]a[/math].
  2. Transpose the quantity [math]\dfrac{c}{a}[/math] to the right side of the equation.
  3. Complete the square by adding [math]\dfrac{b^2}{4a^2}[/math] to both sides of the equation.
  4. Factor the left side and combine the right side.
  5. Extract the square-root of both sides of the equation.
  6. Solve for [math]x[/math] by transporting the quantity [math]\dfrac{b}{2a}[/math] to the right side of the equation.
  7. Combine the right side of the equation to get the quadratic formula.

See the derivation below.

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