11 KiB
| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Triangular Numbers | Obsidian::STEM | 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:
!
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?
!
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.
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%%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 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&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: \large{\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).