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| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Arithmetic Sequence | Obsidian::STEM | algebra::sequence |
|
Overview
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 $a_n = a_{n-1} + d \text{ with } a_0 = aand closed formulaa_n = a + nd.$
%%ANKI Basic What makes a sequence an arithmetic sequence? Back: Each term has the same constant difference from the previous. 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 properties are necessary to characterize an arithmetic sequence? Back: The initial term and the common difference. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What does it mean for the common difference of an arithmetic sequence (a_n) to be d?
Back: a_n - a_{n-1} = d for all valid values of n.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Let (a_n) be an arithmetic sequence. What term refers to the difference between terms?
Back: The common difference.
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 is the "common difference" of an arithmetic sequence named the way it is? Back: It is shared between all successive terms. 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 The term "common difference" is related to what kind of sequence? Back: An arithmetic sequence. 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
Let (a_n)_{n \geq 0}^d be an arithmetic sequence. What is its recursive definition's recurrence relation?
Back: a_n = a_{n-1} + d
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Let (a_n)_{n \geq 0}^d be an arithmetic sequence. What is the $n$th term's closed formula?
Back: a_n = a_0 + nd
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Let (a_n)_{n \geq 0} be an arithmetic sequence. What is the closed formula of \sum_{k=0}^n a_k?
Back: \frac{(a_0 + a_n)(n + 1)}{2}
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Let (a_n)_{n \geq 1} be an arithmetic sequence. What is the closed formula of \sum_{k=1}^n a_k?
Back: \frac{(a_1 + a_n)(n)}{2}
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
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}
Back: The number of terms in the summation.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
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?
Back:
\begin{matrix}
S & = & a_1 & + & a_2 & + & \cdots & + & a_n \\
S & = & a_n & + & a_{n-1} & + & \cdots & + & a_1 \\
\hline
2S & = &(a_1 + a_n) & + & (a_1 + a_n) & + & \cdots & + & (a_1 + a_n)
\end{matrix}$$
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%
%%ANKI
Basic
Interpreted as an arithmetic sequence, what is the common ratio of the triangular numbers?
Back: $1$
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%
%%ANKI
Basic
Let $(a_n)_{n \geq 1}$ be an arithmetic sequence. What does term $2$ correspond to in the following? $$\sum a_k = \frac{(a_1 + a_n)(n)}{2}$$
Back: The double-counting that occurs when adding the summation to itself.
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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END%%
%%ANKI
Basic
Let $(a_n)_{n \geq 1}$ be an arithmetic sequence. How do we visualize the role of term $2$ in the following? $$\sum a_k = \frac{(a_1 + a_n)(n)}{2}$$
Back:
```
* * * * -
* * * - -
* * - - -
* - - - -
```
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).
<!--ID: 1709664600181-->
END%%
%%ANKI
Cloze
You can find the partial sums of {arithmetic} sequences using the "reverse and add" strategy.
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).
<!--ID: 1709666305449-->
END%%
## References
* 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).