# 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.

## 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, … is a geometric progression with
*r*= 2 - 10, -5, 2.5, -1.25, … is a geometric progression with
*r*= -1/2