# 3.3: Finite Difference Calculus

- Page ID
- 13607

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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.

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**-
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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**-
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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.