# Introduction to Progressions

In our study of Mathematics, we will encounter various kinds of **progressions**, which are sequences of numbers with a definite relationship between successive numbers. Let us consider a few examples of progressions.

Consider the following sequence of numbers:

\[ - 3,\, - 1,\,\,1,\,\,3,\,\,5,\,...\]

What do you notice about this sequence? The most important characteristic of this sequence is that the difference between successive terms is constant, and equal to 2. This characteristic enables us to say that the next term in this progression will be 7. This sequence is an example of an **arithmetic progression**. Here is another example of an arithmetic progression:

\[17,\;14,\;11,\;8,\;5,...\]

The difference between successive terms in this progression is –3, and the next term in the sequence will be 2.

Now, consider the following sequence of numbers:

\[2,\,4,\,8,\,16,\,32,...\]

In this sequence, each term is double the previous term. In other words, the ratio of successive terms is constant, and equal to 2. This sequence is an example of a **geometric progression**. In any geometric progression, the ratio of successive terms is constant. Here is another example:

\[1,\,\frac{1}{2},\,\frac{1}{4},\,\frac{1}{8},\,\frac{1}{{16}},...\]

Now, consider the following sequence:

\[\frac{1}{3},\,\frac{1}{5},\,\frac{1}{7},\,\frac{1}{9},\,\frac{1}{{11}},...\]

The important characteristic of this sequence is that the reciprocal of its terms are in arithmetic progression:

\[3,\,\,5,\,\,7,\,\,9,\,\,11,...\]

Such sequences, in which the reciprocal terms are in arithmetic progressions, are called **harmonic progressions**. Here is another example of a harmonic progression:

\[\frac{2}{7},\,\,\frac{2}{{11}},\,\,\frac{2}{{15}},\,\,\frac{2}{{19}},\,\,\frac{2}{{23}},...\]

Verify that its reciprocal terms are in arithmetic progression.