--- 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 polygon. 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). END%% %%ANKI Basic What is a figurate number? Back: Polygonal numbers or generalizations of polygonal numbers to other dimensions. 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). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). 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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%ANKI Basic Algebraically speaking, *what* is the $n$th triangular number? Back: $\sum_{k=1}^n k$. 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). END%% %%ANKI Basic What polygonal sequence is the summation analogue of factorial? Back: The triangular numbers. 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). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%ANKI Basic What closed formula is traditionally used to compute the $n$th triangular number? Back: $\frac{n(n + 1)}{2}$ 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). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%ANKI Basic How do you expand sum $\sum_{k=1}^n k$ 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). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%ANKI Basic Where in Pascal's triangle are the natural numbers embedded? Back: Along the second leftward diagonal: ![[pascals-triangle.webp]] 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). END%% %%ANKI Basic Where in Pascal's triangle are the triangular numbers embedded? Back: Along the third leftward diagonal: ![[pascals-triangle.webp]] 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). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). END%% %%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). 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). * “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).