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Mathematics LibreTexts

3.3: Finite Difference Calculus

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    13607
  • [ "stage:draft", "article:topic", "authorname:thangarajahp", "license:ccbyncsa", "Finite Difference Calculus", "showtoc:yes" ]

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    In this section, we will explore further to the method that we explained at the introduction of Quadratic sequences.

    Example \(\PageIndex{1}\):

    Create a sequence of numbers by finding a relationship between the number of points on the circumference of a circle and the number of regions created by joining the points.

    circle.png

    Number of points on the circle 0 1 2 3 4 5 6 7 8
    Number of regions 1 1 2 4 8 16 31 57 99

    Difference operator

    Notation: Difference operator

    Let \(a_n, n=0,1,2,\cdots\) be a sequence of numbers. Then

    the first difference is defined by \( \Delta a_n = a_{n+1}-a_n, n=1,2,\cdots\). 

    The second difference is defined by \( \Delta^2 a_n =\Delta a_{n+1}- \Delta a_n, n=2,\cdots\). 

    Further, \(k^{th}\) difference is denoted by \(\Delta^k a_n, n=k,\cdots\)

    Let us now consider the sequence of numbers in the example \(\PageIndex{1}\).

    Number of points on the circle 0 1 2 3 4 5 6 7 8
    Number of regions 1 1 2 4 8 16 31 57 99
    \(\Delta a_n\)   0 1 2 4 8 15 26 42
    \(\Delta^2 a_n\)     1 1 2 4 7 11 16
    \(\Delta^3 a_n\)       0 1 2 3 4 5
    \(\Delta^4 a_n\)         1 1 1 1 1

    Since the fourth difference is constant, \(a_n\) should be polynomial of degree \(4\). Let's explore how to find this polynomial.

    Definition

    $$ {n \choose k } =\displaystyle \frac{ n!}{k! (n-k)!} , k \leq n, n \in \mathbb{N} \cup \{0\} .$$

    Theorem \(\PageIndex{1}\)

    $$\Delta {n \choose k } ={n \choose k-1}, k \leq n, n \in \mathbb{N} \cup \{0\} .$$

    Proof

    Add proof here and it will automatically be hidden if you have a "AutoNum" template active on the page.

    Theorem \(\PageIndex{2}\) Newton's formula

    The \(n\) th term of the original sequence is given by $$a_0+ {n \choose 0 } \Delta a_1 +\cdots$$

    Proof

    Add proof here and it will automatically be hidden if you have a "AutoNum" template active on the page.

    If \(n\) th term of the original sequence is linear then the first difference will be a constant. If \(n\) th term of the original sequence is quadratic then the second difference will be a constant. A cubic sequence has the third difference constant. 

    Source

    • Thanks to Olivia Nannan.
    • Reference: Kunin, George B. "The finite difference calculus and applications to the interpolation of sequences." MIT Undergraduate Journal of Mathematics 232.2001 (2001): 101-9.