8.8 KiB
| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| ∆ᵏ-constant Sequence | Obsidian::STEM | algebra::sequence algebra::polynomial |
|
Overview
Let (a_n) be a sequence. We define the k$th differences of (a_n)$ recursively:
- The
0$th differences of(a_n)is(a_n)$. - The
k$th differences of(a_n)is the sequence given by subtracting consecutive terms of the(k-1)st$ differences of(a_n).
A sequence is said to be \Delta^k-constant if the $k$th differences are constant.
The closed formula for a sequence will be a degree
kpolynomial if and only if the sequence is\Delta^k-constant.
This is the discrete analogue to (continuous) derivatives of polynomials.
%%ANKI
Basic
What are the 0$th differences of (a_n)_{n \geq 0}$?
Back: (a_n)_{n \geq 0}
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
What are the 1$st differences of (a_n)_{n \geq 0}$?
Back: (b_n)_{n \geq 1} where b_n = a_n - a_{n - 1}.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
How do we refer to the 1$st differences of (a_n)$ more naturally?
Back: As "the differences of (a_n)."
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
What is the base case of the recursive definition of the k$th differences of (a_n)$?
Back: k = 0
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
What is the recurrence of the recursive definition of the (k + 1)$st differences of (a_n)$?
Back: The $(k + 1)$st differences is the differences of the $k$th differences.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
What does it mean for (a_n) to be \Delta^k-constant?
Back: The k$th differences of (a_n)$ is constant.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
What can be said about the closed formula of a \Delta^k-constant sequence?
Back: It is a polynomial with degree k.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Cloze
We say a sequence is {\Delta^k-constant} when the {$k$th differences is constant}.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
\Delta^k-constant sequences are a discrete analogue to what calculus concept?
Back: Derivatives.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
Tags: calculus
END%%
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Basic
What kind of mathematical expression do \Delta^k-constant sequences relate to?
Back: Polynomials.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
Tags: calculus
END%%
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Basic
How can we prove every quadratic sequence, say (a_n), has arithmetic differences?
Back: By showing a_{n+1} - a_n is linear.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
How can we prove every cubic sequence, say (a_n), has quadratic differences?
Back: By showing a_{n+1} - a_n is quadratic.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Cloze
The closed formula for a sequence will be a {degree k polynomial} if and only if the $k$th differences {is constant}.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
How are arithmetic sequences defined in terms of "\Delta^k-constant"?
Back: A sequence is arithmetic if and only if it is \Delta^1-constant.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
How are geometric sequences defined in terms of "\Delta^k-constant"?
Back: N/A
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
Why can't a geometric sequence be \Delta^k-constant for some k \geq 0?
Back: Because the closed formula of a geometric sequence is not a polynomial.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
Suppose (a_n) is a \Delta^2-constant sequence. What general form describes its closed formula?
Back: an^2 + bn + c
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
Suppose (a_n) is a \Delta^3-constant sequence. What general form describes its closed formula?
Back: an^3 + bn^2 + cn + d
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
Suppose (a_n) is a \Delta^k-constant sequence. Where is k repeated in (a_n)'s closed formula?
Back: At the largest degree of the polynomial.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
What are the terms in a \Delta^0-constant sequences?
Back: c, c, c, \ldots for some constant c.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
How do \Delta^k-constant sequences relate to polynomial fitting?
Back: The closed formula of a \Delta^k-constant sequence is a polynomial we can fit.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
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Basic
How many data points of a \Delta^k-constant sequence are needed to polynomial fit its closed formula?
Back: k + 1
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
Bibliography
- Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.