Arithmetic Progressions

First of all, what is a progression? A progression is a sequence of numbers that follows a specific rule, which can be used to get the nth term of that sequence.

Now, an Arithmetic Progression is a progression of numbers, in which each successive number is obtained by adding a fixed or constant quantity to the previous number. For example the following sequence of numbers is an Arithmetic Progression:

2, 5, 8, 11, 14, ...

In which each number is obtained by adding 3 to the previous number. Here 3 is called the common difference.

Common difference: The constant number which is added to the previous term of an A.P. to obtain the next term is called the common difference. It is represented by the letter d.

If the first term of an arithmetic progression is a, and its common difference be d, then the arithmetic progression is represented by

a, a + d, a + 2d, a + 3d, ... and so on

Formula: The nth term  of an Arithmetic Progression is obtained by the following formula:

Tn = a + (n - 1)d, where

• a represents the first term
• n represents the position of the nth term
• d represents the common difference

Formula: The sum of n terms of an Arithmetic Progression is obtained by:

Sn = n/2(2a + (n-1)d), where

• a represents the first term
• n represents the position of the nth term
• d represents the common difference

Properties of Arithmetic Progressions:

• If three numbers a, b and c are in A.P., then 2b = a + c
• If all terms of an arithmetic progression are added, subtracted, multiplied or divided by the same constant, then the resulting sequence of numbers is also an arithmetic progression.