# Please explain geometric and harmonic progressions?

##### 1 Answer

Arithmetic progression:

Geometric progression:

Harmonic progression:

#### Explanation:

We also have to introduce the arithmetic progression, since the definition of the harmonic progression requires it.

**Arithmetic progression**

An arithmetic progression is a sequence of numbers:

such that the difference between two consecutive numbers is constant:

If we define:

then we have:

and so on, so we can see that the terms of an arithmetic progression can be expressed in the form:

If we consider three consecutive terms we have:

so each term is the arithmetic mean of the terms adjacent to it.

**Geometric progression**

A geometric progression is a sequence of numbers:

such that the ratio between two consecutive numbers is constant:

If we define:

then we have:

and so on, so we can see that the terms of an arithmetic progression can be expressed in the form:

If

so each term is the geometric mean of the terms adjacent to it.

**Harmonic progression**

A harmonic progression is a sequence of numbers:

such that their reciprocal constitute an arithmetic progression

If we define:

then we have:

and so on, so we can see that the terms of an arithmetic progression can be expressed in the form:

If we consider three consecutive terms we have:

so each term is the harmonic mean of the terms adjacent to it.