notebook/notes/algebra/sequences/index.md

title	TARGET DECK	FILE TAGS	tags
Sequences	Obsidian::STEM	algebra::sequence	algebra sequence

Overview

A sequence is an ordered list of numbers. These are usually represented as a closed formula or a recursive definition.

%%ANKI Basic What is a sequence? Back: An ordered list of values. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic How does sequence 0, 1, 2, \ldots differ from set \mathbb{N}? Back: Order matters in the former. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic How is sequence a_1, a_2, \ldots more compactly denoted? Back: (a_n)_{n \geq 1} Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What is the first index of (a_n)_{n \geq 1}? Back: 1 Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What terms make up the expanded form of sequence (a_n)_{n \geq 0}? Back: a_0, a_1, a_2, \ldots Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What term refers to the subscripts in sequence a_0, a_1, a_2, \ldots? Back: Indices. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic How do functions and sequences relate to one another? Back: Every sequence is equivalent to a function mapping index to value. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic How is sequence a_0, a_1, a_2, \ldots interpreted as a function? Back: As a function that assigns n \mapsto a_n. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What two methods allow exact specification of a sequence? Back: Closed formulas and recursive definitions. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What is a closed formula for sequence e.g. (a_n)_{n \geq 0}? Back: A formula that can be computed using a fixed number of "well-known" standard operations, independent of n. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

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%%ANKI Cloze {n!} is a closed formula even though equivalent expression {\Pi_{k=1}^n k} is not. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

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%%ANKI Basic Why is \Pi_{k=1}^n k not considered a closed formula? Back: The number of "well-known" standard operations (i.e. multiplications) scale with n. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

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%%ANKI Basic Why is n! considered a closed formula? Back: Because the factorial operation is considered a "well-known" standard operation. Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

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%%ANKI Basic What two parts make up a recursive definition of a sequence? Back: The recurrence relation and an initial condition. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What distinguishes a recurrence relation and a recursive definition? Back: A recursive definition includes a recurrence relation along with an initial condition. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic How is sequence (a_n)_{n \geq 0} = 1, 2, 4, 8, \ldots represented as a closed formula? Back: a_n = 2^n Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic How is sequence (a_n)_{n \geq 0} = 1, 2, 4, 8, \ldots represented as a recursive definition? Back: a_n = 2 \cdot a_{n-1} with a_0 = 1. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Cloze It is harder to find {terms of a sequence} with a recursive definition than closed formula. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic Which is considered easier to find: closed formulas or recursive definitions? Back: Recursive definitions. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What general strategy is typically employed when finding a closed formula for a sequence? Back: First find a recursive definition. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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Recurrence Relations

To solve a recurrence relation means to find a closed form for the relation (with respect to initial conditions).

%%ANKI Basic What is the recurrence relation for the Fibonacci sequence? Back: F_n = F_{n-1} + F_{n-2} Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What does it mean to solve a recurrence relation? Back: To find a closed formula satisfying the relation and initial conditions. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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Telescoping

Telescoping refers to the property of summations in which consecutive terms cancel out. We can use telescoping to solve recurrences of form a_n = a_{n-1} + f(n) by noticing that: \begin{align*} a_1 - a_0 & = f(1) \ a_2 - a_1 & = f(2) \ \vdots \ a_n - a_{n-1} & = f(n) \ \hline a_n - a_0 & = \sum_{k=1}^n f(n) \end{align*}

%%ANKI Basic What does it mean for a sum to be telescoping? Back: Pairs of consecutive terms in the summation cancel each other out. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What imagery is invoked by the term "telescoping" with respect to a sum? Back: A collapsing telescope. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What summands typically remain after evaluating a telescoping sum? Back: The first and the last. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What is the result of the following after observing telescoping? (2 - 1) + (3 - 2) + \cdots + (100 - 99) + (101 - 100) Back: -1 + 101 Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic How is the following rewritten to highlight telescoping? (2 - 1) + (3 - 2) + \cdots + (100 - 99) + (101 - 100) Back: (-1 + 2) + (-2 + 3) + \cdots + (-99 + 100) + (-100 + 101) Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What is the result of the following? \sum_{n=1}^N (a_n - a_{n-1}) Back: a_N - a_0 Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What property is used to quickly verify the following identity? \sum_{n=1}^N (a_n - a_{n-1}) = a_N - a_0$$ Back: This is a telescoping sum. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic Schematically show how telescoping can be used to solve a_n = a_{n-1} + f(n). Back: \begin{align*} a_1 - a_0 & = f(1) \ & \vdots \ a_n - a_{n-1} & = f(n) \ \hline a_n - a_0 & = \sum_{k=1}^n f(k) \end{align*}$$Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What is the closed formula of recurrence a_n = a_{n-1} + f(n)? Back: a_n = a_0 + \sum_{k=1}^n f(k) Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What summation property can be used to derive the closed formula of a_n = a_{n-1} + f(n)? Back: Telescoping. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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Iteration

Iteration refers to the expansion of terms, starting at the initial conditions, in the hope of discovering a pattern. It is more general than #Telescoping is. Consider a_n = a_{n-1} + f(n) again. We solve with iteration like so: \begin{align*} a_1 & = a_0 + f(1) \ a_2 & = (a_0 + f(1)) + f(2) \ \vdots \ a_n & = (\cdots(a_0 + f(1)) + f(2)) + \cdots) + f(n) \ \hline a_n & = a_0 + \sum_{k=1}^n f(n) \end{align*}

%%ANKI Basic What does it mean to solve a recurrence relation using iteration? Back: Repeatedly expand terms, starting at the initial conditions, to discover a pattern. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic What is the result of "iterating" the following recursive definition twice? $a_n = 3a_{n-1} + 2$ Back: \begin{align*} a_1 & = 3a_0 + 2 \ a_2 & = 3(3a_0 + 2) + 2 \end{align*} Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic Schematically show how iteration can be used to solve a_n = a_{n-1} + f(n). Back: \begin{align*} a_1 & = a_0 + f(1) \ & \vdots \ a_n & = (\cdots((a_0 + f(1)) + f(2)) + \cdots) + f(n) \ \hline a_n & = a_0 + \sum_{k=1}^n f(k) \end{align*} Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic How fast does the number of terms grow when iterating a_n = 3a_{n-1} + 2? Back: Linearly. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic How fast does the number of terms grow when iterating a_n = 2a_{n-1} + 3a_{n-2}? Back: Exponentially (the number of terms double each iteration). Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic Why should you avoid using iteration to solve a_n = 2a_{n-1} + 3a_{n-2}? Back: The number of terms grows unwieldy very quickly. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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%%ANKI Basic When solving recurrences, is telescoping or iteration a more general technique? Back: Iteration. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

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Characteristic Roots

When encountering linear homogeneous recurrence relations with constant coefficients, we can use the characteristic root technique to solve. We demonstrate with a quadratic characteristic polynomial, though this technique generalizes to higher-order polynomials as well.

Given recurrence relation a_n + \alpha a_{n-1} + \beta a_{n-2} = 0, the characteristic polynomial is r^2 + \alpha r + \beta. If r_1 and r_2 are distinct roots of the characteristic polynomial, then the solution to the recurrence relation is $a_n = ar_1^n + br_2^n$ where a and b are determined by the initial conditions. If the characteristic polynomial only has one root r, the solution is instead $a_n = ar^n + bnr^n$

%%ANKI Basic The characteristic root technique only works when solving what kind of recurrence relation? Back: Linear homogeneous recurrence relations with constant coefficients. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic What does "linear" refer to in "linear homogeneous recurrence relations with constant coefficients"? Back: The recurrence relation is a linear combination of previous terms. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic What does "homogeneous" refer to in "linear homogeneous recurrence relations with constant coefficients"? Back: Every term in the relation is a multiple of previous terms. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic Why isn't a_n = 2a_{n-1} + 3a_{n-2} a linear homogeneous recurrence relation with constant coefficients? Back: N/A. It is. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic Why isn't a_n = 2a_{n-1} \cdot 3a_{n-2} a linear homogeneous recurrence relation with constant coefficients? Back: It is non-linear. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic Why isn't a_n = 2a_{n-1} + 3 a linear homogeneous recurrence relation with constant coefficients? Back: It is non-homogeneous. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic Why isn't a_n = 2a_{n-1} + na_{n-2} a linear homogeneous recurrence relation with constant coefficients? Back: It has a nonconstant coefficient. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic What is the most famous linear homoegeneous recurrence relation with constant coefficients? Back: The Fibonacci sequence's recurrence relation. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic Why might we guess the solution to e.g. a_n = a_{n-1} + 6a_{n-2} is geometric? Back: Every step of iteration multiplies a previous iteration by 6. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic How is a_n = a_{n-1} + 6a_{n-2} factored to yield its characteristic polynomial? Back: r^{n-2}(r^2 - r - 6) = 0 Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic What is the characteristic equation of a_n = a_{n-1} + 6a_{n-2}? Back: r^2 - r - 6 = 0 Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic What is the characteristic polynomial of a_n = a_{n-1} + 6a_{n-2}? Back: r^2 - r - 6 Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic What is the characteristic polynomial of a_n + \alpha a_{n-1} + \beta a_{n-2} = 0? Back: r^2 + \alpha r + \beta Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic What is the characteristic equation of a_n + \alpha a_{n-1} + \beta a_{n-2} = 0? Back: r^2 + \alpha r + \beta = 0 Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic What is the characteristic equation of a_n = \alpha a_{n-1} + \beta a_{n-2}? Back: r^2 - \alpha r - \beta = 0 Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic Given recurrence a_n = \alpha a_{n-1} + \beta a_{n-2}, what guess is used to derive the concept of a characteristic polynomial? Back: The guessing of a geometric solution, e.g. r^n. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic What distinguishes the characteristic polynomial from the characteristic equation of a recurrence relation? Back: The latter sets the characteristic polynomial equal to 0. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic Using the quadratic characteristic root technique, what distinguishes the form of the closed solution? Back: The number of distinct roots of the characteristic polynomial. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic Suppose the characteristic polynomial of a_n = \alpha a_{n-1} + \beta a_{n-2} has distinct roots r_1 and r_2. What is its solution? Back: a_n = ar_1^n + br_2^n where a and b are determined by initial conditions. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic Suppose the characteristic polynomial of a_n = \alpha a_{n-1} + \beta a_{n-2} has single root r. What is its solution? Back: a_n = ar^n + bnr^n where a and b are determined by initial conditions. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic How many distinct roots does the characteristic polynomial of the following have? $a_n = \alpha a_{n-1} + \beta a_{n-2} = ar^n + bnr^n$ Back: One. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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%%ANKI Basic How many distinct roots does the characteristic polynomial of the following have? $a_n = \alpha a_{n-1} + \beta a_{n-2} = ar_1^n + br_2^n$ Back: Two. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algebra::polynomial

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Bibliography

• Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
• Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed (Reading, Mass: Addison-Wesley, 1994).