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 an = d (n  1) + c, where d is the common difference between consecutive terms, and c = a1. An arithmetic sequence can also be defined recursively by the formulas a1 = c, an+1 = an + 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 a1 and the value of the last term an. Then, the sum of the first n terms of the arithmetic sequence is Sn = n(). To use the second method, you must know the value of the first term a1 and the common difference d. Then, the sum of the first n terms of an arithmetic sequence is Sn = na1 + (dn  d ).
2  +  5  +  8  +  11  +  14  =  40 
14  +  11  +  8  +  5  +  2  =  40 


16  +  16  +  16  +  16  +  16  =  80 
Computation of the sum 2 + 5 + 8 + 11 + 14. When the sequence is reversed and added to itself term by term, the resulting sequence has a single repeated value in it, equal to the sum of the first and last numbers (2 + 14 = 16). Thus 16 × 5 = 80 is twice the sum.
The sum of the members of a finite arithmetic progression is called an arithmetic series. For example, consider the sum:
 $\{\backslash displaystyle\; 2+5+8+11+14\}$
This sum can be found quickly by taking the number n of terms being added (here 5), multiplying by the sum of the first and last number in the progression (here 2 + 14 = 16), and dividing by 2:
 $\{\backslash displaystyle\; \{\backslash frac\; \{n(a\_\{1\}+a\_\{n\})\}\{2\}\}\}$
In the case above, this gives the equation:
 $\{\backslash displaystyle\; 2+5+8+11+14=\{\backslash frac\; \{5(2+14)\}\{2\}\}=\{\backslash frac\; \{5\backslash times\; 16\}\{2\}\}=40.\}$
This formula works for any real numbers
$\{\backslash displaystyle\; a\_\{1\}\}$
and
$\{\backslash displaystyle\; a\_\{n\}\}$
. For example:
 $\{\backslash displaystyle\; \backslash left(\{\backslash frac\; \{3\}\{2\}\}\backslash right)+\backslash left(\{\backslash frac\; \{1\}\{2\}\}\backslash right)+\{\backslash frac\; \{1\}\{2\}\}=\{\backslash frac\; \{3\backslash left(\{\backslash frac\; \{3\}\{2\}\}+\{\backslash frac\; \{1\}\{2\}\}\backslash right)\}\{2\}\}=\{\backslash frac\; \{3\}\{2\}\}.\}$
Sequence – a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts at one
finite sequence – has a last term
infinite sequence – continues without stopping