arithmetic progression

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 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
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