163 lines
7.1 KiB
Markdown
163 lines
7.1 KiB
Markdown
---
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title: Arithmetic Sequence
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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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An **arithmetic sequence** $(a_n)_{n \geq 0}^d$ is a sequence in which each term differs by a constant $d$. Given initial term $a$, it has recursive definition $$a_n = a_{n-1} + d \text{ with } a_0 = a$$ and closed formula $$a_n = a + nd.$$
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%%ANKI
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Basic
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What makes a sequence an *arithmetic* sequence?
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Back: Each term has the same constant difference from the previous.
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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: 1709664600159-->
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END%%
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%%ANKI
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Basic
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What two properties are necessary to characterize an arithmetic sequence?
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Back: The initial term and the common difference.
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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: 1709664600161-->
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END%%
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%%ANKI
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Basic
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What does it mean for the common difference of an arithmetic sequence $(a_n)$ to be $d$?
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Back: $a_n - a_{n-1} = d$ for all valid values of $n$.
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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: 1709664600162-->
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END%%
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%%ANKI
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Basic
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Let $(a_n)$ be an arithmetic sequence. What term refers to the difference between terms?
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Back: The common difference.
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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: 1709664600164-->
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END%%
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%%ANKI
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Basic
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Why is the "common difference" of an arithmetic sequence named the way it is?
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Back: It is shared between all successive terms.
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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: 1709664600166-->
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END%%
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%%ANKI
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Basic
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The term "common difference" is related to what kind of sequence?
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Back: An arithmetic sequence.
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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: 1709664600167-->
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END%%
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%%ANKI
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Basic
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Let $(a_n)_{n \geq 0}^d$ be an arithmetic sequence. What is its recursive definition's recurrence relation?
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Back: $a_n = a_{n-1} + d$
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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: 1709664600169-->
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END%%
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%%ANKI
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Basic
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Let $(a_n)_{n \geq 0}^d$ be an arithmetic sequence. What is the $n$th term's closed formula?
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Back: $a_n = a_0 + nd$
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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: 1709664600170-->
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END%%
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%%ANKI
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Basic
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Let $(a_n)_{n \geq 0}$ be an arithmetic sequence. What is the closed formula of $\sum_{k=0}^n a_k$?
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Back: $$\frac{(a_0 + a_n)(n + 1)}{2}$$
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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: 1709664600172-->
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END%%
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%%ANKI
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Basic
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Let $(a_n)_{n \geq 1}$ be an arithmetic sequence. What is the closed formula of $\sum_{k=1}^n a_k$?
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Back: $$\frac{(a_1 + a_n)(n)}{2}$$
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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: 1709664600173-->
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END%%
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%%ANKI
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Basic
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Let $(a_n)_{n \geq 1}$ be an arithmetic sequence. What does term $n$ correspond to in the following? $$\sum a_k = \frac{(a_1 + a_n)(n)}{2}$$
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Back: The number of terms in the summation.
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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: 1709664600175-->
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END%%
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%%ANKI
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Basic
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Let $(a_n)_{n \geq 1}$ be an arithmetic sequence. How do you expand $\sum_{k=1}^n a_k$ to derive its closed formula?
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Back:
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$$\begin{matrix}
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S & = & a_1 & + & a_2 & + & \cdots & + & a_n \\
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S & = & a_n & + & a_{n-1} & + & \cdots & + & a_1 \\
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\hline
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2S & = &(a_1 + a_n) & + & (a_1 + a_n) & + & \cdots & + & (a_1 + a_n)
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\end{matrix}$$
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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: 1709664600176-->
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END%%
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%%ANKI
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Basic
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Interpreted as an arithmetic sequence, what is the common ratio of the triangular numbers?
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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: 1709664600178-->
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END%%
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%%ANKI
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Basic
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Let $(a_n)_{n \geq 1}$ be an arithmetic sequence. What does term $2$ correspond to in the following? $$\sum a_k = \frac{(a_1 + a_n)(n)}{2}$$
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Back: The double-counting that occurs when adding the summation to itself.
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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: 1709664600179-->
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END%%
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%%ANKI
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Basic
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Let $(a_n)_{n \geq 1}$ be an arithmetic sequence. How do we visualize the role of term $2$ in the following? $$\sum a_k = \frac{(a_1 + a_n)(n)}{2}$$
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Back:
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```
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* * * * -
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* * * - -
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* * - - -
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* - - - -
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```
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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: 1709664600181-->
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END%%
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%%ANKI
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Cloze
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You can find the partial sums of {arithmetic} sequences using the "reverse and add" strategy.
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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: 1709666305449-->
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END%%
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%%ANKI
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Basic
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Why is a sequence of partial sums named the way it is?
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Back: Each term is found by adding a finite number of infinite terms.
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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: 1713580109297-->
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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). |