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# 3.3: Finite Difference Calculus

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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 Number of regions 0 1 2 3 4 5 6 7 8 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.