Sequences and Series By: Olivia, Jon, Jordan, and Jaymie

 Sequence- a list of numbers that usually forms a pattern  Each number in the list is a  All sequences can be or  Sequences have a general rule: a n =  Recursive sequences  Must give you rule for finding terms based on previous terms  Must give you a 1 or a 2  Previous term is labeled or  Factorial  If n is a positive integer, then n!=

 Series- the sum of terms in a sequence  Can be or  Just like sequences, but instead of there are  Summation notation (aka sigma notation)  i is called the n is called the i=1 is called the “the as goes from to of ”

 Arithmetic Sequence- has a common difference between consecutive terms  Common difference- number you to each term to get the term  General Rule a n = a 1 + (n – 1)d d= difference between terms  Sum of a Finite Arithmetic Sequence  S n = n is how many a 1 is the a n is the

 Geometric Sequence- have a common ratio between terms  Common Ratio- the number you multiply each term by to get the term  General Rule n is r is  Sum of a Finite Geometric Sequence   Infinite Geometric Series  If |r|< 1, then

 Mathematical Induction- a mathematical proof about statements involving positive integers  Principle of Mathematical Induction  Let P n be a statement involving a integer n If P 1 is true then assume P k is true If P k is true it is implied that P k+1 is also true If all conditions apply then P n is true for all positive integers

 If all differences in the sequence are equal,then the sequence has a a n =an+b  If all the differences are, but the differences are, then the sequence has a a n =an ² +bn+c