Arithmetic Progressions
We have seen that an arithmetic progression is a sequence of numbers in which the difference between successive terms is constant. From now on, we will abbreviate arithmetic progression as AP. Here are some more examples of APs:
\[\begin{array}{l}6,\,\,13,\,\,20,\,\,27,\,\,34,...\\91,\,\,81,\,\,71,\,\,61,\,\,51,...\\\pi ,\,\,2\pi ,\,\,3\pi ,\,\,4\pi ,\,\,5\pi ,...\\  \sqrt 3 ,\,\,  2\sqrt 3 ,\,\,  3\sqrt 3 ,\,\,  4\sqrt 3 ,\,\,  5\sqrt 3 ,...\end{array}\]
Two parameters are sufficient to define an AP completely:

First term: The name is selfexplanatory. This is the first number of the sequence. In the first series above, the first term is 6. For an arbitrary AP, the first term will generally be denoted by a.

Common difference: This is the difference between successive terms, and is in general denoted by d. In the first series above, d is equal to 7.
Let us see some more examples of APs, with their first terms and common differences specified explicitly:
5, 7, 9, 11, 13,...  a = 5, d = 2 
101, 81, 61, 41, 21,...  a = 101, d =  20 
0,  \(\pi \),  2\(\pi \),  3\(\pi \),  4\(\pi \),...  a = 0, d =  \(\pi \) 
7.3, 7.4, 7.5, 7.6, 7.7,...  a = 7.3, d = 0.1 
We note that the terms of an AP can take on any real values. Also, the common difference can take any real value – positive, negative or zero. In the case that d is equal to 0, we will have a constant AP, like the one below:
\[4,\,\,4,\,\,4,\,\,4,\,\,4,...\]
In terms of a and d, the terms of an AP can be written as follows:
\[a,\,\,a + d,\,\,a + 2d,\,\,a + 3d,\,\,a + 4d,...\]