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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Solving Recurrence Relations

%%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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%%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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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).