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Some of the content of this guide was modeled after a guide originally created by the Openstax and has been adapted for the GPRC Learning Commons in November 2020. This work is licensed under a Creative Commons BY 4.0 International License

Arithmetic Sequences

An arithmetic series is a series in which the difference between a term and next term is a constant. This constant is called the common difference. The general form of an arithmetic sequence is given by:

in the above equation, is the first term of the sequence and is the common difference. Note that each term of the sequence differs from the pervious term by . The last term of the sequence, denoted by  is given by:



Example: Is the following sequence an arithmetic sequence?


If a sequence is an arithmetic sequence, the difference between successive terms has to be a constant. We have to find the difference between the first term and the second term, the second term and the term term etc. If the difference is a constant, then the sequence is an arithmetic sequence. 

Since the difference is NOT a constant, the sequence is not an arithmetic sequence.


Example: For an arithmetic sequence, . Find .

For an arithmetic sequence, the general term is , therefore for we get:


Example: The general term of an arithmetic sequence is and also  Find the number of terms of this arithmetic sequence.
We use the formula for the general term of an arithmetic sequence and substitute the given values in the formula:

therefore, we get:



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