176 lines
7.5 KiB
Markdown
176 lines
7.5 KiB
Markdown
---
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title: Sequences
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TARGET DECK: Obsidian::STEM
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FILE TAGS: algebra::sequence
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tags:
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- algebra
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- sequence
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---
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## Overview
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A **sequence** is an ordered list of numbers. These are usually represented as a **closed formula** or a **recursive definition**.
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%%ANKI
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Basic
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What is a sequence?
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Back: An ordered list of values.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343193-->
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END%%
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%%ANKI
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Basic
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How does sequence $0, 1, 2, \ldots$ differ from set $\mathbb{N}$?
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Back: Order matters in the former.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343198-->
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END%%
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%%ANKI
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Basic
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How is sequence $a_1, a_2, \ldots$ more compactly denoted?
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Back: $(a_n)_{n \geq 1}$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343201-->
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END%%
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%%ANKI
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Basic
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What is the first index of $(a_n)_{n \geq 1}$?
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Back: $1$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343204-->
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END%%
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%%ANKI
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Basic
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What terms make up the expanded form of sequence $(a_n)_{n \geq 0}$?
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Back: $a_0, a_1, a_2, \ldots$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343207-->
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END%%
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%%ANKI
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Basic
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What term refers to the subscripts in sequence $a_0, a_1, a_2, \ldots$?
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Back: Indices.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343211-->
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END%%
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%%ANKI
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Basic
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How do functions and sequences relate to one another?
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Back: Every sequence is equivalent to a function mapping index to value.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343214-->
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END%%
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%%ANKI
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Basic
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How is sequence $a_0, a_1, a_2, \ldots$ interpreted as a function?
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Back: As a function that assigns $n \mapsto a_n$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343216-->
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END%%
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%%ANKI
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Basic
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What two methods allow exact specification of a sequence?
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Back: Closed formulas and recursive definitions.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343220-->
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END%%
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%%ANKI
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Basic
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What is a closed formula for sequence e.g. $(a_n)_{n \geq 0}$?
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Back: A formula that can be computed using a fixed number of "well-known" standard operations, independent of $n$.
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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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<!--ID: 1709302343222-->
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END%%
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%%ANKI
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Cloze
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{$n!$} is a closed formula even though equivalent expression {$\Pi_{k=1}^n k$} is not.
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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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<!--ID: 1709302343225-->
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END%%
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%%ANKI
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Basic
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Why is $\Pi_{k=1}^n k$ not considered a closed formula?
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Back: The number of "well-known" standard operations (i.e. multiplications) scale with $n$.
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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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<!--ID: 1709302343228-->
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END%%
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%%ANKI
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Basic
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Why is $n!$ considered a closed formula?
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Back: Because the factorial operation is considered a "well-known" standard operation.
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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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<!--ID: 1709302343231-->
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END%%
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%%ANKI
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Basic
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What two parts make up a recursive definition of a sequence?
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Back: The recurrence relation and an initial condition.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343234-->
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END%%
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%%ANKI
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Basic
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What distinguishes a recurrence relation and a recursive definition?
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Back: A recursive definition includes a recurrence relation along with an initial condition.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343238-->
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END%%
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%%ANKI
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Basic
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How is sequence $(a_n)_{n \geq 0} = 1, 2, 4, 8, \ldots$ represented as a closed formula?
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Back: $a_n = 2^n$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343242-->
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END%%
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%%ANKI
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Basic
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How is sequence $(a_n)_{n \geq 0} = 1, 2, 4, 8, \ldots$ represented as a recursive definition?
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Back: $a_n = 2 \cdot a_{n-1}$ with $a_0 = 1$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343245-->
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END%%
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%%ANKI
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Cloze
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It is harder to find {terms of a sequence} with a recursive definition than closed formula.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343249-->
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END%%
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%%ANKI
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Basic
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Which is considered easier to find: closed formulas or recursive definitions?
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Back: Recursive definitions.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343252-->
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END%%
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%%ANKI
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Basic
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What general strategy is typically employed when finding a closed formula for a sequence?
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Back: First find a recursive definition.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709302343255-->
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END%%
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## Bibliography
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* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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* 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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