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
Below is the derivation of this relationship.
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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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