An arithmetic sequence has a common difference (\(d\)) between consecutive terms.

An arithmetic series is the sum of the terms in an arithmetic sequence.

Formulae

The \(n^{th}\) term of an arithmetic sequence is given by:

\(u_n=a+(n-1)d\)

where \(u_n\) is the \(n^{th}\) term,
\(a\) is the first term, and
\(d\) is the common difference.

The sum of the first \(n\) terms in an arithmetic series is given by:

\(S_n=\dfrac{n}{2}(2a+(n-1)d)\)

where \(S_n\) is the sum of the first \(n\) terms,
\(n\) is the number of terms being added up,
\(a\) is the first term, and
\(d\) is the common difference.

If the last term is known, then this can be written as:

\(S_n=\dfrac{n}{2}(a+l)\)

where \(S_n\) is the sum of the first \(n\) terms,
\(n\) is the number of terms being added up,
\(a\) is the first term, and
\(l\) is the last term.

Use of arithmetic sequences and series in modelling

For example, saving schemes where each year a constant amount is paid into an account, e.g. simple interest.