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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