Neither Arithmetic nor Geometric Sequences
When analyzing sequences in Math 1, students are often asked to determine if a sequence is arithmetic, geometric, or neither. The use of the word neither is meant to say that a sequence is not arithmetic, nor geometric. However, I often have many of my students interpret the word “neither” to mean none, such that they believe patterns that are not arithmetic, nor geometric may have no pattern or algebraic representation. In response to this issue, I would discuss and show my students examples of a variety of sequences that behave in different ways. For example, the Fibonacci sequence is not arithmetic or geometric, but is defined by a simple recursive equation. I also know that some students work with this sequence in our introductory computer science class.
It is not until the Math 2 course that students are shown that there are also sequences that can be modeled using quadratic expressions. However, in Math 2, the discussion is brief and relies on telling students that a sequence is quadratic if the second difference is constant. Textbooks show how to find the second difference by drawing a diagram,3 and then ask students to write an equation to represent the terms in the quadratic sequence. Yet, the examples given rely on other properties of quadratics, such as knowing both intercepts, to find equations. Moreover, the discussion and examples in the text do not provide students with any methods to find equations for the general quadratic sequence.
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