An arithmetic sequence is a sequence in which the difference between
each consecutive term is constant. An arithmetic sequence can be defined by
an explicit formula in which
*a*
_{n} = *d* (*n* - 1) + *c*
, where
*d*
is the common
difference between consecutive terms, and
*c* = *a*
_{1}
. An arithmetic sequence can
also be defined recursively by the formulas
*a*
_{1} = *c*, *a*
_{n+1} = *a*
_{n} + *d*
, in
which
*d*
is again the common difference between consecutive terms, and
*c*
is a
constant.

The sum of an infinite arithmetic sequence is either
∞
, if
*d* > 0
, or
- ∞
, if
*d* < 0
.

There are two ways to find the sum of a finite arithmetic sequence. To use the
first method, you must know the value of the first term
*a*
_{1}
and the value of
the last term
*a*
_{n}
. Then, the sum of the first
*n*
terms of the arithmetic
sequence is
*S*
_{n} = *n*()
. To use the second method, you must
know the value of the first term
*a*
_{1}
and the common difference
*d*
. Then, the
sum of the first
*n*
terms of an arithmetic sequence is
*S*
_{n} = *na*
_{1} + (*dn* - *d* )
.