Common Core Algebra II.Unit 5.Lesson 1.Sequences

Summary

In this Algebra 2 lesson, Kirk Wier explores the concept of sequences, reviewing both arithmetic and recursive types. He explains sequences as functions with positive integer domains and demonstrates how to define and calculate terms using both explicit formulas and recursive relations. The lesson includes examples like odd integers, the Fibonacci sequence, and recursive sequences where each term is defined based on the previous one. Students learn to generate terms, identify patterns, and convert sequences into algebraic formulas. The lesson also covers graphing sequences as sets of discrete points and concludes with practical strategies for solving sequence-related problems.

Takeaways

• 😀 A sequence is a function where the domain is the set of positive integers (1, 2, 3, etc.), and its output represents a list of ordered numbers.
• 😀 Sequences can be defined explicitly with formulas, such as `a(n) = 2n - 1`, to generate terms like 1, 3, 5, 7, etc.
• 😀 A recursive sequence depends on the previous term, such as `f(n) = f(n-1) + 5`, where each term is derived by adding a constant to the previous term.
• 😀 The Fibonacci sequence is a well-known recursive sequence, where each term is the sum of the two preceding terms (`f(n) = f(n-1) + f(n-2)`), starting with 1, 1.
• 😀 In recursive sequences, it's important to know the starting value, as each subsequent term depends on the previous one.
• 😀 To find a term in a recursive sequence, it’s helpful to understand the pattern and number of times an operation (like adding 5 or multiplying by 2) is repeated.
• 😀 The explicit formula for a sequence provides a direct method to calculate any term, while the recursive formula relies on previously computed terms.
• 😀 In some sequences, like `a(n) = 2n + 1`, the value of each term increases linearly with `n`, which can be graphed as a set of points on the coordinate plane.
• 😀 Graphing sequences on a coordinate plane helps visualize how the sequence grows, showing both the x-values (positions) and y-values (terms).
• 😀 To convert a sequence into an algebraic formula, you need to find the pattern, whether it involves addition, multiplication, or other operations (e.g., `a(n) = 2^n`).

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