# JEE Main & Advanced Mathematics Arithmetic Progressions Definition

## Definition

Category : JEE Main & Advanced

A sequence of numbers $<{{t}_{n}}>$ is said to be in arithmetic progression (A.P.) when the difference ${{t}_{n}}-{{t}_{n-1}}$ is a constant for all n Î N. This constant is called the common difference of the A.P. and is usually denoted by the letter d.

If $'a'$ is the first term and $'d'$ the common difference, then an A.P. can be represented as $a,\,a+d,a+2d,\,a+3d,........$

Example : 2, 7, 12, 17, 22, …… is an A.P. whose first term is 2 and common difference 5.

Algorithm to determine whether a sequence is an A.P. or not.

Step I: Obtain ${{a}_{n}}$ (the ${{n}^{th}}$ term of the sequence).

Step II: Replace $n$ by $n-1$ in ${{a}_{n}}$ to get ${{a}_{n-1}}$.

Step III: Calculate ${{a}_{n}}-{{a}_{n-1}}$.

If ${{a}_{n}}-{{a}_{n-1}}$ is independent from $n,$ the given sequence is an A.P. otherwise it is not an A.P.

$\therefore$ ${{t}_{n}}=An+B$ represents the ${{n}^{th}}$ term of an A.P. with common difference A.

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