geometric 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 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]
Syndicate content