Sequences and Series
Arithmetic Sequences
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 )
.
