geometric progression

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}
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