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# 3.4: Finite Difference Calculus


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:

Let $$a_n, n=0,1,2,\cdots$$ be a sequence of numbers. Then the first difference is defined by $$\Delta a_n = a_{n+1}-a_n, n=0,1,2,\cdots$$.

The second difference is defined by $$\Delta (\Delta a_n)= \Delta^2 a_n =\Delta a_{n+1}- \Delta a_n, n=0,1,2,\cdots$$.

Further, $$k^{th}$$ difference is denoted by $$\Delta^k a_n, n=k,\cdots$$. We shall denote that $$\Delta^0 a_n=a_n$$.

Example $$\PageIndex{2}$$

Let $$a_n=n^2$$, for all $$n \in \mathbb{N}$$.  Show that $$\Delta^2 a_n=2$$.

Solution

$$\Delta a_n =(n+1)^2-n^2= 2n+1,$$

$$\Delta^2 a_n=(2(n+1)+1)-(2n+1)= 2.$$

Rule

Let $$k \in \mathbb{Z_+}$$. Let $$a_n=n^k$$, for all $$n \in \mathbb{N}$$.  Then $$\Delta^k a_n=k!$$.

Example $$\PageIndex{3}$$

Let $$a_n=c^n$$, for all $$n \in \mathbb{N}$$.  Show that $$\Delta a_n=(c-1) a_n$$.

Solution

$$\Delta a_n= c^{n+1}-c^{n}= c^n(c-1) = (c-1) a_n$$.

Below are some properties of the difference operator.

Theorem $$\PageIndex{1}$$: Difference operator is a Linear Operator

Let $$a_n$$ and $$b_n$$ be sequences, and let $$c$$ be any number. Then

1.  $$\Delta (a_n+b_n)= \Delta a_n+ \Delta b_n$$, and
2.  $$\Delta ( c a_n) =c \Delta a_n$$.
Proof

Very simple calculation.

Definition: Falling Powers

The falling power is denoted by $$x^{\underline{m}}$$ and it is defined by $$x (x-1)(x-2)(x-3) \cdots (x-m+1)$$, with $$x^{\underline{0}}=1$$.

Falling powers are useful in the difference calculus because of the following property:

Theorem $$\PageIndex{2}$$

$$\Delta n^{\underline{m}} =m \,n^{\underline{m-1}}$$.

Proof

$$\Delta n^{\underline{m}}= (n+1)^{\underline{m}}-n^{\underline{m}}$$

$$=(n+1) n (n-1)(n-2)(n-3) \cdots (n-m+2)- n (n-1)(n-2)(n-3) \cdots (n-m+1)$$

$$=n (n-1)(n-2)(n-3) \cdots (n-m+2) ( (n+1)-(n-m+1))$$

$$= n (n-1)(n-2)(n-3) \cdots (n-m+2) \,m$$

$$=m\, n^{\underline{m-1}}$$.

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 $$\PageIndex{2}$$

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

Proof

Note that $$\displaystyle {n \choose k }= \frac{1}{k!} n^{\underline{k}}.$$

Consider $$\Delta {n \choose k } = \Delta ( \frac{1}{k!} n^{\underline{k}})$$

$$= \frac{1}{k!} \Delta( n^{\underline{k}})$$

$$= \frac{k}{k!} n^{\underline{(k-1)}}$$

$$= \frac{1}{(k-1)!} n^{\underline{(k-1)}}$$

$$= {n \choose k-1}$$.

We can also see this by considering Pascal’s triangle.

Theorem $$\PageIndex{3}$$ Newton's formula

Let $$a_0, a_1, a_2, \cdots$$ be sequence of numbers such that  $$\Delta^{k+1} a_0=0.$$ Then the  $$n$$ th term of the original sequence is given by

$$a_n= \frac{1}{k!}(\Delta^k a_0)n^{\underline{k}} + \cdots+ a_0= a_0+ {n \choose 0 } \Delta a_1 +{n \choose 2 } \Delta^2 a_2+ \cdots + {n \choose k } \Delta^k a_k.$$

Proof

Exercise.

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 for the diagram.
• 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.
• Reference: Samson, D. (2006). Number patterns, cautionary tales and finite differences. Learning and Teaching Mathematics2006(3), 3-8.