notebook/notes/algebra/sequences/triangular-numbers.md

---
title: Triangular Numbers
TARGET DECK: Obsidian::STEM
FILE TAGS: algebra::sequence
tags:
  - algebra
  - sequence
---

## Overview

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 $$0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \ldots$$

%%ANKI
Basic
What is a polygonal number?
Back: A number of pebbles that can be arranged into the shape of a regular filled polygon.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
What is a figurate number?
Back: Polygonal numbers and their generalizations to other dimensions.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
What are considered the simplest polygonal numbers?
Back: The triangular numbers.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
How do polygonal numbers relate to figurate numbers?
Back: Polygonal numbers are a subset of the figurate numbers.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
What is a gnomon?
Back: The "piece" added to a figurate number to transform it to the next larger one.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
What shape do gnomons associated with triangular numbers take on?
Back: Lines.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
How are gnomons of the triangular numbers visualized?
Back:
![[triangular-gnomon.png]]
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
What general term refers to the highlighted portion of pebbles in the following?
![[triangular-gnomon.png]]
Back: Gnomons.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
The triangular numbers correspond to what kind of triangles?
Back: Equilateral triangles.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
How do the triangular numbers correspond to *equilateral* triangles?
Back:
![[triangular-gnomon.png]]
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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END%%

%%ANKI
Basic
What is the first triangular *and* square number?
Back: $36$
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
What are the first five triangular numbers $(T_n)_{n \geq 0}$?
Back: $0, 1, 3, 6, 10$
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
How is triangular number $10$ graphically depicted?
Back:
```
   *
  * *
 * * *
* * * *
```
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
How is the $n$th triangular number written as a summation?
Back: $\sum_{k=1}^n k$
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
What polygonal sequence is the summation analog of factorial?
Back: The triangular numbers.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
What notation does Knuth introduce to denote the $n$th triangular number?
Back: $n?$
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
What name does Knuth give the LHS of $n? = \sum_{k=1}^n k$?
Back: The termial.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Cloze
The {1:term}ial is to {2:$n?$} as the {2:factor}ial is to {1:$n!$}.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
What closed formula is traditionally used to compute the $n$th triangular number?
Back: $\large{\frac{n(n + 1)}{2}}$
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
What is the recurrence relation in the recursive definition of triangular numbers $(T_n)_{n \geq 0}$?
Back: $T_n = T_{n-1} + n$
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
What is the initial condition(s) in the recursive definition of triangular numbers $(T_n)_{n \geq 0}$?
Back: $T_0 = 0$
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
How do you expand $\sum_{k=1}^n k$ to derive closed formula $\frac{n(n + 1)}{2}$?
Back:
$$\begin{matrix}
1 & + & 2 & + & \cdots & + & n \\
n & + & (n - 1) & + & \cdots & + & 1 \\
\hline
(n + 1) & + & (n + 1) & + & \cdots & + & (n + 1)
\end{matrix}$$
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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%%ANKI
Basic
What combinatorial closed formula is used to compute the $n$th triangular number?
Back: $\binom{n + 1}{2}$
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
What is the combinatorial explanation as to why the $n$th triangular number is $\binom{n + 1}{2}$?
Back: $\sum_{k=1}^n k$ is the number of ways distinct pairs can be made from $n + 1$ objects.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
Where in Pascal's triangle are the natural numbers embedded?
Back: Along the second leftward diagonal:
![[pascals-triangle.png]]
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
Where in Pascal's triangle are the triangular numbers embedded?
Back: Along the third leftward diagonal:
![[pascals-triangle.png]]
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
What polygonal number is $k$ equal to after the following `for` loops?
```c
int k = 0;
for (int i = 1; i <= n; ++i) {
  k += i;
}
```
Back: The $n$th triangular number.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
Why is $n(n + 1)$ geometrically significant w.r.t. the $n$th triangular number?
Back: $2 \cdot T_n$ is the number of units in an $n \times (n + 1)$ rectangle, e.g.
```
* * * * -
* * * - -
* * - - -
* - - - -
```
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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%%ANKI
Basic
The triangular numbers is the sequence of partial sums of what sequence?
Back: $a_n = n$
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).
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## Bibliography

* 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).
* “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).