notebook/notes/algebra/sequences/delta-constant.md

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title TARGET DECK FILE TAGS tags
∆ᵏ-constant Sequence Obsidian::STEM algebra::sequence algebra::polynomial
algebra
sequence

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 k polynomial 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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

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%%ANKI 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

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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%%ANKI 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.

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Bibliography