Sequences and Series
Lessons in this chapter
About Sequences and Series
Sequences and Series builds from basic sequence vocabulary through to summing infinitely many terms.
Defining and Using Sequences and Series introduces the basic vocabulary and notation for a sequence and its running sum, before Analyzing Arithmetic Sequences and Series and Analyzing Geometric Sequences and Series cover the two core sequence types and their sum formulas.
Finding Sums of Infinite Geometric Series shows that an infinite geometric series can still add up to a finite number, under the right condition on its common ratio.
Using Recursive Rules with Sequences closes the chapter by defining a sequence in terms of its own previous terms instead of a direct formula.
Each of the five lessons, from Defining and Using Sequences and Series through Finding Sums of Infinite Geometric Series to Using Recursive Rules with Sequences, opens as a worksheet worked in the page. The practice problems are scored as you answer, so mixing up a common difference with a common ratio shows up straight away. Free to practice online.
Frequently asked questions
Defining and Using Sequences and Series separates an ordered list of terms from the sum of those terms. Analyzing Arithmetic Sequences and Series covers the constant difference and its sum formula, Analyzing Geometric Sequences and Series does the same for the constant ratio, and Finding Sums of Infinite Geometric Series shows when an endless sum still settles on a finite value. Using Recursive Rules with Sequences closes by defining each term from the ones before it. This sits in Functions (F) in the high school Common Core, with the geometric series sum formula in Algebra (A).
A sequence is an ordered list of numbers; a series is the sum of a sequence's terms: the two are related but describe different things.
An arithmetic sequence adds the same constant difference between consecutive terms; its series has a specific sum formula based on the first term, last term, and number of terms.
A geometric sequence multiplies by the same constant ratio between consecutive terms; its series has its own sum formula based on the first term, ratio, and number of terms.
An infinite geometric series only has a finite sum when its common ratio is between -1 and 1: otherwise the terms grow instead of shrinking toward zero, and the sum diverges.
A recursive rule defines each term of a sequence using the term(s) before it, rather than a direct formula based on the term's position.