132 lines
5.3 KiB
Markdown
132 lines
5.3 KiB
Markdown
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---
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title: Square 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 **square numbers** $(s_n)_{n \geq 0}$ is $n^2$. The first few terms are $$0, 1, 4, 9, 16, 25, 36, 49, 64, \ldots$$
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%%ANKI
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Basic
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What shape do gnomons associated with square numbers take on?
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Back: L-shapes.
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Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731).
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<!--ID: 1709422558613-->
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END%%
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%%ANKI
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Basic
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How are gnomons of the square numbers visualized?
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Back:
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![[square-gnomon.png]]
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Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731).
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<!--ID: 1709422558619-->
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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 different colored segments in the following?
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![[square-gnomon.png]]
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Back: Gnomons.
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Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731).
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<!--ID: 1709422558622-->
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END%%
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%%ANKI
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Basic
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What are the first five square numbers $(s_n)_{n \geq 0}$?
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Back: $0, 1, 4, 9, 16$
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Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731).
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<!--ID: 1709422558625-->
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END%%
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%%ANKI
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Basic
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How is square number $16$ 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: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731).
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<!--ID: 1709422558628-->
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END%%
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%%ANKI
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Basic
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What closed formula is used to find the $n$th square number?
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Back: $n^2$
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Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731).
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<!--ID: 1709422558631-->
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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 square numbers $(s_n)_{n \geq 0}$?
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Back: $s_n = s_{n-1} + (2n - 1)$
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Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731).
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<!--ID: 1709422558634-->
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END%%
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%%ANKI
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What is the initial condition(s) in the recursive definition of square numbers $(s_n)_{n \geq 0}$?
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Back: $s_0 = 0$
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Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731).
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END%%
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%%ANKI
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Basic
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How is the $n$th square number $s_n$ represented with sigma notation?
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Back: $s_n = \sum_{k=1}^n (2k - 1)$
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Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731).
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<!--ID: 1709422558638-->
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END%%
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%%ANKI
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Basic
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Which polygonal numbers are the "next" generalization of triangular numbers?
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Back: The square 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: 1709419325898-->
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END%%
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%%ANKI
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Cloze
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The sum of {1:natural numbers} is to {2:triangular numbers} whereas the sum of {2:odd natural numbers} is to {1:square numbers}.
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Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731).
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<!--ID: 1709422558641-->
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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 += 2 * i - 1;
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}
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```
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Back: The $n$th square number.
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Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731).
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<!--ID: 1709422558645-->
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END%%
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%%ANKI
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Basic
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*Why* are square numbers a sum of odd numbers?
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Back: The gnomon of a square number is twice the width of the previous square, plus the corner.
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Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731).
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<!--ID: 1709422558648-->
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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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* “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731).
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