diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 7a2dd4d..c477cad 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -231,14 +231,16 @@ "algebra/sequences/index.md": "e5a7cdfbcb61709ce2963c4b5e53a8f2", "_journal/2024-03-02.md": "08c3cae1df0079293b47e1e9556f1ce1", "_journal/2024-03/2024-03-01.md": "70da812300f284df72718dd32fc39322", - "algebra/sequences/triangular-numbers.md": "18925b0ecae151c3e6d38bc018c632c4", + "algebra/sequences/triangular-numbers.md": "c6626d8aa86776b6ce794c3551862d5a", "algebra/sequences/square-numbers.md": "886fb22fb8dbfffdd2cd233558ea3424", "_journal/2024-03-03.md": "c4977a3778ed227b768c3f9ad5512670", "_journal/2024-03/2024-03-02.md": "8136792b0ee6e08232e4f60c88d461d2", "_journal/2024-03-04.md": "9ec052061e7a613ff877a4488576e82f", "_journal/2024-03/2024-03-03.md": "64e2f17b4d57a6bd42a3d1b7f2851b83", - "_journal/2024-03-05.md": "e049435d23c401cc28076a4e1edd1c37", - "_journal/2024-03/2024-03-04.md": "4948d90a08af2cff58c629c9a2e11ee4" + "_journal/2024-03-05.md": "a285ac3e48e335c50c42ee20b0cb0472", + "_journal/2024-03/2024-03-04.md": "4948d90a08af2cff58c629c9a2e11ee4", + "algebra/sequences/geometric.md": "53936ec392b3b714bd4a9bdb4554b582", + "algebra/sequences/arithmetic.md": "80381ca0f2b3b9a1c155c597a7dea75a" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-03-05.md b/notes/_journal/2024-03-05.md index a17ca51..703688c 100644 --- a/notes/_journal/2024-03-05.md +++ b/notes/_journal/2024-03-05.md @@ -10,4 +10,9 @@ title: "2024-03-05" - [ ] Interview Prep (1 Practice Problem) - [ ] Log Work Hours (Max 3 hours) -* Start on arithmetic/geometric sequences and their sums. \ No newline at end of file +* 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. \ No newline at end of file diff --git a/notes/algebra/sequences/arithmetic.md b/notes/algebra/sequences/arithmetic.md new file mode 100644 index 0000000..a3840b7 --- /dev/null +++ b/notes/algebra/sequences/arithmetic.md @@ -0,0 +1,155 @@ +--- +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). \ No newline at end of file diff --git a/notes/algebra/sequences/geometric.md b/notes/algebra/sequences/geometric.md new file mode 100644 index 0000000..f1db2a6 --- /dev/null +++ b/notes/algebra/sequences/geometric.md @@ -0,0 +1,147 @@ +--- +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). \ No newline at end of file diff --git a/notes/algebra/sequences/triangular-numbers.md b/notes/algebra/sequences/triangular-numbers.md index 420bd41..450d179 100644 --- a/notes/algebra/sequences/triangular-numbers.md +++ b/notes/algebra/sequences/triangular-numbers.md @@ -180,7 +180,7 @@ END%% %%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 \\