260 lines
11 KiB
Markdown
260 lines
11 KiB
Markdown
---
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title: Triangular Numbers
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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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The $n$th term of the **triangular numbers** $(T_n)_{n \geq 0}$ is the sum of whole numbers $\sum_{k=0}^n k$. The first few terms are $$0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \ldots$$
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%%ANKI
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Basic
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What is a polygonal number?
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Back: A number of pebbles that can be arranged into the shape of a regular polygon.
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325851-->
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END%%
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%%ANKI
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Basic
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What is a figurate number?
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Back: Polygonal numbers or generalizations of polygonal numbers to other dimensions.
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325856-->
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END%%
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%%ANKI
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Basic
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What are considered the simplest polygonal numbers?
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Back: The triangular numbers.
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325859-->
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END%%
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%%ANKI
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Basic
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How do polygonal numbers relate to figurate numbers?
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Back: Polygonal numbers are a subset of the figurate numbers.
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325862-->
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END%%
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%%ANKI
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Basic
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What is a gnomon?
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Back: The "piece" added to a figurate number to transform it to the next larger one.
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325865-->
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END%%
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%%ANKI
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Basic
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What shape do gnomons associated with triangular numbers take on?
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Back: Lines.
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325874-->
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END%%
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%%ANKI
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Basic
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How are gnomons of the triangular numbers visualized?
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Back:
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![[triangular-gnomon.png]]
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325878-->
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END%%
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%%ANKI
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Basic
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What general term refers to the highlighted portion of pebbles in the following?
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![[triangular-gnomon.png]]
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Back: Gnomons.
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325883-->
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END%%
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%%ANKI
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Basic
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The triangular numbers correspond to what kind of triangles?
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Back: Equilateral triangles.
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325887-->
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END%%
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%%ANKI
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Basic
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What is the first triangular *and* square number?
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Back: $36$
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325891-->
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END%%
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%%ANKI
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Basic
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What are the first five triangular numbers $(T_n)_{n \geq 0}$?
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Back: $0, 1, 3, 6, 10$
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325904-->
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END%%
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%%ANKI
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Basic
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How is triangular number $10$ graphically depicted?
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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: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325909-->
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END%%
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%%ANKI
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Basic
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Algebraically speaking, *what* is the $n$th triangular number?
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Back: $\sum_{k=1}^n k$.
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325914-->
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END%%
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%%ANKI
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Basic
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What polygonal sequence is the summation analogue of factorial?
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Back: The triangular numbers.
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325918-->
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END%%
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%%ANKI
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Basic
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What notation does Knuth introduce to denote the $n$th triangular number?
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Back: $n?$
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325922-->
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END%%
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%%ANKI
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Basic
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What name does Knuth give the LHS of $n? = \sum_{k=1}^n k$?
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Back: The termial.
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325927-->
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END%%
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%%ANKI
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Cloze
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The {1:term}ial is to {2:$n?$} as the {2:factor}ial is to {1:$n!$}.
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325931-->
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END%%
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%%ANKI
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Basic
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What closed formula is traditionally used to compute the $n$th triangular number?
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Back: $\frac{n(n + 1)}{2}$
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325936-->
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END%%
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%%ANKI
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Basic
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What is the recurrence relation in the recursive definition of triangular numbers $(T_n)_{n \geq 0}$?
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Back: $T_n = T_{n-1} + n$
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709422558652-->
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END%%
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%%ANKI
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Basic
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What is the initial condition(s) in the recursive definition of triangular numbers $(T_n)_{n \geq 0}$?
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Back: $T_0 = 0$
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709422558656-->
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END%%
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%%ANKI
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Basic
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How do you expand sum $\sum_{k=1}^n k$ to derive closed formula $\frac{n(n + 1)}{2}$?
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Back:
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$$\begin{matrix}
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1 & + & 2 & + & \cdots & + & n \\
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n & + & (n - 1) & + & \cdots & + & 1 \\
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\hline
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(n + 1) & + & (n + 1) & + & \cdots & + & (n + 1)
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\end{matrix}$$
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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: 1709419325942-->
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END%%
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%%ANKI
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Basic
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What combinatorial closed formula is used to compute the $n$th triangular number?
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Back: $\binom{n + 1}{2}$
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325949-->
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END%%
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%%ANKI
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Basic
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What is the combinatorial explanation as to why the $n$th triangular number is $\binom{n + 1}{2}$?
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Back: $\sum_{k=1}^n k$ is the number of ways distinct pairs can be made from $n + 1$ objects.
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325956-->
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END%%
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%%ANKI
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Basic
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Where in Pascal's triangle are the natural numbers embedded?
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Back: Along the second leftward diagonal:
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![[pascals-triangle.webp]]
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325963-->
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END%%
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%%ANKI
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Basic
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Where in Pascal's triangle are the triangular numbers embedded?
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Back: Along the third leftward diagonal:
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![[pascals-triangle.webp]]
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325970-->
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END%%
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%%ANKI
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Basic
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What polygonal number is $k$ equal to after the following `for` loops?
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```c
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int k = 0;
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for (int i = 1; i <= n; ++i) {
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k += i;
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}
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```
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Back: The $n$th triangular number.
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Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325976-->
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END%%
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%%ANKI
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Basic
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Why is $n(n + 1)$ geometrically significant w.r.t. the $n$th triangular number?
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Back: $2 \cdot T_n$ is the number of units in an $n \times (n + 1)$ rectangle, e.g.
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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: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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<!--ID: 1709419325981-->
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END%%
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## References
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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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* “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). |