8.1: Sequences Craters of the Moon National Park, Idaho Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon, Siena College Photo by Vickie Kelly, 2008

nth term A sequence is a list of numbers written in an explicit order. Any real-valued function with domain a subset of the positive integers is a sequence. If the domain is finite, then the sequence is a finite sequence. In calculus, we will mostly be concerned with infinite sequences.

A sequence is defined explicitly if there is a formula that allows you to find individual terms independently. Example: To find the 100th term, plug 100 in for n:

A sequence is defined recursively if there is a formula that relates an to previous terms. Example: We find each term by looking at the term or terms before it: You have to keep going this way until you get the term you need.

An arithmetic sequence has a common difference between terms. Example: Arithmetic sequences can be defined recursively: or explicitly:

An geometric sequence has a common ratio between terms. Example: Geometric sequences can be defined recursively: or explicitly:

Example: If the second term of a geometric sequence is 6 and the fifth term is -48, find an explicit rule for the nth term.

You can determine if a sequence converges by finding the limit as n approaches infinity. Does converge? The sequence converges and its limit is 2.

p Absolute Value Theorem for Sequences If the absolute values of the terms of a sequence converge to zero, then the sequence converges to zero. p