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

END%%

%%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&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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

END%%

%%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&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.

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.

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.

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.

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.

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.

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.

END%%

%%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).
<!--ID: 1709419325942-->
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).
<!--ID: 1709419325949-->
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).
<!--ID: 1709419325956-->
END%%

%%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325963-->
END%%

%%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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325970-->
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).
<!--ID: 1709419325976-->
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).
<!--ID: 1709419325981-->
END%%

%%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).
<!--ID: 1713580109292-->
END%%

## 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&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).