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Arithmetic and geometric sequences.

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Joshua Potter 2024-03-05 12:28:11 -07:00
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notes/_journal/2024-03-05.md
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- [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)
* Start on arithmetic/geometric sequences and their sums.
* Arithmetic and geometric sequences in "Discrete Mathematics: An Open Introduction".
* Completed chapter exercises 2.2.
* TODO
* Finish up tree objects and commit objects.
* 101weiqi
* 2 interview problems.

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notes/algebra/sequences/arithmetic.md Normal file
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---
title: Arithmetic Sequence
TARGET DECK: Obsidian::STEM
FILE TAGS: algebra::sequence
tags:
- 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 $$a_n = a_{n-1} + d \text{ with } a_0 = a$$ and closed formula $$a_n = a + nd.$$
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709664600159-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709664600161-->
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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709664600162-->
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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709664600164-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709664600166-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709664600167-->
END%%
%%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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709664600169-->
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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709664600170-->
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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709664600172-->
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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709664600173-->
END%%
%%ANKI
Basic
Let $(a_n)_{n \geq 1}$ be an arithmetic sequence. What does term $n$ correspond to in the following? $$\sum_{k=1}^n a_k = \frac{(a_1 + a_n)(n)}{2}$$
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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709664600175-->
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).
<!--ID: 1709664600176-->
END%%
%%ANKI
Basic
The triangular numbers belong to what larger class of sequences?
Back: Arithmetic sequences.
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: 1709664600178-->
END%%
%%ANKI
Basic
Let $(a_n)_{n \geq 1}$ be an arithmetic sequence. What does term $2$ correspond to in the following? $$\sum_{k=1}^n a_k = \frac{(a_1 + a_n)(n)}{2}$$
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).
<!--ID: 1709664600179-->
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? $$\sum_{k=1}^n a_k = \frac{(a_1 + a_n)(n)}{2}$$
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).

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---
title: Geometric Sequence
TARGET DECK: Obsidian::STEM
FILE TAGS: algebra::sequence
tags:
- algebra
- sequence
---
## Overview
A **geometric sequence** $(a_n)_{n \geq 0}^r$ is a sequence in which each term differs by a constant multiple of $r$. Given initial term $a$, it has recursive definition $$a_n = ra_{n-1} \text{ with } a_0 = a$$ and closed formula $$a_n = ar^n.$$
%%ANKI
Basic
What makes a sequence a *geometric* sequence?
Back: Each term is the same constant multiple from the previous.
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: 1709664600144-->
END%%
%%ANKI
Basic
What two properties are necessary to characterize a geometric sequence?
Back: The initial term and the common ratio.
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: 1709664600147-->
END%%
%%ANKI
Basic
What does it mean for the common ratio of a geometric sequence $(a_n)$ to be $r$?
Back: $a_n = a_{n-1} \cdot r$ 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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709664600149-->
END%%
%%ANKI
Basic
Let $(a_n)$ be a geometric sequence. What term refers to the ratio between terms?
Back: The common ratio.
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: 1709664600151-->
END%%
%%ANKI
Basic
Why is the "common ratio" of a geometric 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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709664600152-->
END%%
%%ANKI
Basic
The term "common ratio" is related to what kind of sequence?
Back: A geometric sequence.
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: 1709664600153-->
END%%
%%ANKI
Basic
Let $(a_n)_{n \geq 0}^r$ be a geometric sequence. What is its recursive definition's recurrence relation?
Back: $a_n = a_{n-1} \cdot r$
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: 1709664600155-->
END%%
%%ANKI
Basic
Let $(a_n)_{n \geq 0}^r$ be a geometric sequence. What is the $n$th term's closed formula?
Back: $a_n = a_0 \cdot r^n$
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: 1709664600156-->
END%%
%%ANKI
Cloze
{1:Arithmetic} sequences are characterized by the common {2:difference}. {2:Geometric} sequences are characterized by the common {1:ratio}.
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: 1709664600158-->
END%%
%%ANKI
Basic
Let $(a_n)_{n \geq 0}^r$ be a geometric sequence. What is the closed formula of $\sum_{k=0}^n a_k$?
Back: $$\frac{a_0(1 - r^{n+1})}{1 - r}$$
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: 1709666305433-->
END%%
%%ANKI
Basic
Let $(a_n)_{n \geq 1}^r$ be a geometric sequence. What is the closed formula of $\sum_{k=1}^n a_k$?
Back: $$\frac{a_1(1 - r^n)}{1 - r}$$
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: 1709666305436-->
END%%
%%ANKI
Basic
Let $(a_n)_{n \geq 1}^r$ be a geometric sequence. What does term $n$ correspond to in the following? $$\sum_{k=1}^n a_k = \frac{a_1(1 - r^n)}{1 - r}$$
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](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
<!--ID: 1709666305438-->
END%%
%%ANKI
Basic
Let $(a_n)_{n \geq 1}^r$ be a geometric sequence. How do you expand $\sum_{k=1}^n a_k$ to derive its closed formula?
Back:
$$\begin{matrix}
S & = & a_1r^0 & + & a_1r^1 & + & \cdots & + & a_1r^{n-1} & + & 0 \\
rS & = & 0 & + & a_1r^1 & + & \cdots & + & a_1r^{n-1} & + & a_1r^n \\
\hline
S - rS & = & a_1r^0 & & & & & & & - & a_1r^n
\end{matrix}$$
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).
<!--ID: 1709666305441-->
END%%
%%ANKI
Basic
Let $(a_n)_{n \geq 1}^r$ be a geometric sequence. How is term $1 - r$ derived in the following? $$\sum_{k=1}^n a_k = \frac{a_1(1 - r^n)}{1 - r}$$
Back: Given $S = \sum_{k=1}^n a_k$, by factoring out $S$ from $S - rS$.
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: 1709666356524-->
END%%
%%ANKI
Cloze
You can find the partial sums of {geometric} sequences using the "multiply and subtract" 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: 1709666305444-->
END%%
%%ANKI
Cloze
{1:Reverse} and {1:add} arithmetic sequences. {2:Multiply} and {2:subtract} geometric sequences.
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: 1709666305447-->
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).

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%%ANKI
Basic
How do you expand sum $\sum_{k=1}^n k$ to derive closed formula $\frac{n(n + 1)}{2}$?
How do you expand $\sum_{k=1}^n k$ to derive closed formula $\frac{n(n + 1)}{2}$?
Back:
$$\begin{matrix}
1 & + & 2 & + & \cdots & + & n \\