algebra

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

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

Let AM = arithmetic mean, GM = geometric mean, and HM = 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, r of geometric progression. Each term therefore in geometric progression is found by multiplying the previous one by r.

Eaxamples of GP:

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

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 x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} give the roots of a quadratic equation which may be real or imaginary. The \pm sign in the radical indicates that x_1 = \dfrac{-b + \sqrt{b^2-4ac}}{2a} and x_2 = \dfrac{-b - \sqrt{b^2-4ac}}{2a} where x_1 and x_2 are the roots of the quadratic equation ax^2 + bx + c = 0. The sum of roots x_1 + x_2 and the product of roots x_1 \, x_2 are common to problems involving quadratic equation.

Derivation of Quadratic Formula

The roots of a quadratic equation ax^2 + bx + c = 0 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 ax^2 + bx + c = 0 by a.
  2. Transpose the quantity \dfrac{c}{a} to the right side of the equation.
  3. Complete the square by adding \dfrac{b^2}{4a^2} 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 x by transporting the quantity \dfrac{b}{2a} 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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