Mathematical Sequences
Each number in a sequence is a term and can be identified by its position in a sequence. Given a sequence, students are expected to extend patterns to find subsequent terms, create an algebraic representation of the problem by writing an explicit formula in terms of n for the n-th term in the sequence, and use their explicit formulas to further analyze the sequence. This unit will discuss arithmetic and geometric sequences using the sequence notation, an, while for quadratic sequences, we will be using function notation f(n). Either representation is acceptable, and I chose to use different notations because the sequence notation appears in the common course exams used at our school for Math 1, while function terminology is introduced after arithmetic and geometric sequences. Note that the terms arithmetic and linear are often used synonymously, as are geometric and exponential.
A difference sequence is a sequence found by taking the difference between two successive terms of an original sequence. In general, any sequence is determined by its first term and its difference sequence. This follows because the nth term, an, of the original sequence is the sum of the i) first term of the original sequence and ii) the first n-1 terms of the difference sequence. At the high school level, this fact is often provided without proof or further formal elaboration, as the proof requires induction. Al Cuoco provides a thorough explanation and justification through examples and proof in the text Mathematical Connections.2 Finding the difference sequence is useful because it may be simpler than the original sequence. We can also take the difference sequence of an existing difference sequence, which is called the second difference. Using the same argument as seen with the first difference, it would then follow that the original sequence is determined by i) its first term; ii) the first term of its first difference sequence and iii) its second difference sequence. And so forth, for further difference sequences. As demonstrated in the sections below, I plan to discuss with my students these facts and demonstrate how to find difference sequences and writing linear sequences in terms of the first term and their difference sequence through multiple examples.
This unit uses the fact that any sequence is determined by its first term and its difference sequence to highlight the connection that the general result about reconstructing a sequence from its first term and its difference sequence can be followed by study of sequences for which the difference sequence is constant, i.e., all terms have the same value. In particular, for linear sequences, the first difference sequence is constant, and vice versa. With my Math 1 students, I would discuss the derivation of the formula for the nth term of a linear sequence with a constant difference, by showing multiple examples of these types of sequences. Below, I will show how I would build an equation for an arithmetic sequence.
For quadratic sequences, the second difference sequence is constant. I would show my students how to verify sequences as quadratic by finding the second differences of a sequence. For my pre-calculus students, I would show to my students through subsequent applications of the same argument, the general theorem for differences which states that a constant nth difference implies a nth degree polynomial will model the sequence. The general theorem is also stated as a fact for high school students to use by Cuoco, with no formal proof provided for students, but a proof is provided for teachers as enrichment in Mathematical Connections. For exponential sequences, the difference sequence is also exponential, with the same ratio between terms. Difference sequences are not as useful for studying exponential sequences since the result is a sequence with the same level of difficulty/complication. I would have my students work with sequences without constant differences so they understand that difference sequences are only a tool to help us analyze and classify the type of sequence.
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