# Arithmetic sequence

An arithmetic sequence is a type of sequence in which the difference between each consecutive term in the sequence is constant. For example, the difference between each term in the following sequence is 3:

2, 5, 8, 11, 14, 17, 20...

To expand the above arithmetic sequence, starting at the first term, 2, add 3 to determine each consecutive term. This is simple for the first few terms, but using this method to determine terms further along in the sequence gets tedious very quickly. Fortunately, the nth term of an arithmetic sequence can be determined using

a_{n} = a_{1} + (n - 1)d

where a_{n} is the n^{th} term, a_{1} is the initial term, and d is the constant difference between each term. Using the above sequence, the formula becomes:

a_{n} = 2 + 3n - 3 = 3n - 1

Therefore, the 100th term of this sequence is:

a_{100} = 3(100) - 1 = 299

This formula allows us to determine the nth term of any arithmetic sequence.

## Arithmetic sequence vs arithmetic series

An arithmetic series is the sum of a finite part of an arithmetic sequence. For example, 2 + 5 + 8 = 15 is an arithmetic series of the first three terms in the sequence above. The sum of a finite arithmetic sequence can be found using the following formula,

where n is the number of terms in the sequence, a_{1} is the first term in the sequence, and a_{n} is the n^{th} term, and d is the constant difference between each term.

Example

Find the sum of the first 7 terms in the arithmetic sequence 1, 7, 13, 19, 25, ...

First determine the rest of the first 7 terms. The difference between each consecutive term in the sequence is 6. Since we only need to determine 2 more terms, it is simpler to just add 6 to the last given term and the term after that. The first 7 terms are:

1, 7, 13, 19, 25, 31, 37

The sum of these terms is: