A. Definition

sequences  an ordered list of numbers.

terms  the numbers in the sequence.(a variable with a subscript number gives the term place in the sequence such as means the 7^{th} term)

general term  denoted by a_{n} and is the n^{th} term.

arithmetic sequence  a sequence where the difference “d” between consecutive terms is constant.
 4, 9, 14, 19, 24, … is an arithmetic sequence because there is a common difference of 5.
 17, 14, 11, 8, 5, … is an arithmetic sequence because there is a common difference of 3.

rule  an equation that allows you to find any term in the sequence.
B . The Rule for an Arithmetic Sequence: a_{n} = a_{1} + (n  1)d
 a_{n} is the n^{th} term of the sequence.
 a_{1} is the first term of the sequence.
 n is the number of terms in the sequence.
 d is the common difference.
 Use only the a_{1} and d values to write the rule.
C. Writing a Rule When You Are Only Given the Arithmetic Sequence
 Determine the a_{1} and d values.
 Substitute the a_{1} and d values into a_{n} = a_{1} + (n  1)d.
 Simplify the equation.


D. Writing a Rule When You Know Some Term In the Arithmetic Sequence and the Common Difference.
 Find a_{1} by substituting the given information into a_{n} = a_{1} + (n  1)d.
 Substitute the a_{1} and d values only into a_{n} = a_{1} + (n  1)d.
 Simplify the equation.


E. Writing a Rule When You Only Know Two Terms in the Arithmetic Sequence.
 Write a system of equations.

Eq. 1: substitute the largest n into a_{n} = a_{1} + (n  1)d.

Eq. 2: substitute the smallest n into a_{n} = a_{1} + (n  1)d.
 Simplify each equation.
 Subtract the equations (Eq. 1  Eq. 2) to find d.
 Substitute the value of d into Eq. 2 (the "smallest equation") to find a_{1}.
 Substitute the values of a_{1} and d into a_{n} = a_{1} + (n  1)d.
 Simplify the equation.


