Progressions

Arithmetic Progression

An arithmetic progression is a sequence in which each term after the first is formed by adding a fixed amount, called the common difference, to the preceding term.

If a is the first term, d is the common difference, and n is the number of terms an arithmetic progression, the successive terms are

a, a + d, a + 2d, a + 3d,….a + (n-1)d

Thus, the last term (or nth term) l is given by

l= a + (n - 1)d

The sum S of the n terms of this progression is given by

 or 

Arithmetic Means

The terms between the first and last terms of an arithmetic progression are called arithmetic means between these two terms. Thus to insert k arithmetic means between two numbers is to form an arithmetic progression of (k+2) terms having the two given numbers as the first and the last terms.

Geometric Progression

A geometric progression is a sequence in which each term after the first is formed by multiplying the preceding term by a fixed number, called the common ratio.

If a is the first term, r is the common ratio, and n is the number of terms, the geometric progression is:

a, a∙r, a∙r², ….. a∙r^n-1

Thus, the last ( or nth ) term l is given by l = a∙r^n-1.

The sum S of the first n terms of the geometric progression is given by:  or

Geometric Means

The terms between the first and the last terms of a geometric progression are called geometric means between two terms. Thus, to insert k geometric means between two numbers is to form a geometric progression of k+2 terms having the two given numbers as the first and last terms.