Introduction
In mathematics, a sequence is defined as an ordered list of numbers. Each number in a sequence is called a term. The length of a sequence can vary. Finite sequences, are sequences that stop after a certain number of terms. Infinite sequences do not stop and are assumed to continue indefinitely. For example, the sequence 2,4,6,8,10 is a finite sequence with five terms, while 2,4,6,8,10,… is an infinite sequence. Many potentially infinite sequences are formed according to simple rules, and such sequences are also often referred to as “patterns”.
At the high school level, students are introduced to three main special types of sequences: arithmetic (linear), geometric (exponential), and quadratic. This curriculum unit discusses each of these three types in the sections below. When given a few terms in a sequence, students are expected to classify the sequence type, as being potentially one of the three special ones, or “other”. They are also asked to identify key characteristics of the sequence, and to use these to write both recursive equations and explicit equations for an, the nth term in the sequence. Note that some texts may use the term rule in place of equation. Many of my students are able to observe patterns represented in the sequences, but struggle in taking those observations and translate translating them into algebraic representations.
In textbooks, formulas are frequently just given to students based on specific types of sequences. This unit will look at how the common difference, second difference, or common ratio between consecutive terms of a sequence can be used to determine a sequence’s recursive and explicit equations. In addition to tabular models, geometric representations of sequences will be used as an alternative approach to represent sequences, leading to algebraic representations. Overall, this unit serves as a supplement to enhance my existing textbook resources, to better support students in any course that works with linear, quadratic, and exponential sequences and functions.
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