Sequences in Probability Theory

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Sequences are building blocks for patterns and series. They are often used to solve real-world problems across different fields, like computer science, physics, and economics.

In this chapter, we will go through the definition and types of sequences, along with a set of examples for a better understanding of the concept.

What is a Sequence?

In simple terms, a sequence is an ordered list of numbers or terms. Each element in a sequence is known as a "term". A sequence can be finite (ending after a certain number of terms) or infinite (continuing indefinitely). Unlike sets, the elements must be put in order, or in other words, the order of terms in a sequence matters.

For example, the sequence 0, 1, 2, 3, … represents all non-negative integers in a specific order. The term a₀ represents the first element, followed by a₁, a₂, and so on.

Defining a Sequence

Sequences are typically defined in two ways −

• Using a closed formula, or
• A recursive definition.

Closed Formula

A closed formula shows us how to find a term in a sequence using a direct calculation based on the term's position. It shows the sequence like a function.

Here, the input is the position number n and the output is the term a_n. For example, the closed formula for the sequence a_n = 2n defines each term as twice the position number n. Thus,

• a₀ = 0,
• a₁ = 2,
• a₂ = 4, and so on.

The advantage of a closed formula is that it provides a straightforward way to find any term in the sequence without knowing the previous terms.

Recursive Definition

A sequence can also be defined using recursion. Here a recursive definition means recurrence relation. This defines each term in a sequence by relating it to previous terms.

For example, in the Fibonacci sequence −

$$\mathrm{F_n = F_{n-1} + F_{n-2}}$$

where F₀ = 0 and F₁ = 1. Here, each term is the sum of the previous two terms. The recursive definitions are useful when a pattern is difficult to capture with a closed formula. But can be defined by the relation of each term to its predecessors.

Types of Sequences

Sequences vary greatly in structure. Some commonly used types in discrete mathematics include the arithmetic sequences, geometric sequences, Fibonacci sequences, and other specialized sequences.

Arithmetic Sequence

In an arithmetic sequence, each term is obtained by adding a fixed number. This number is known as the common difference. It is added to the previous term.

For example, the sequence 3, 6, 9, 12, … has a common difference of 3, so the general formula is −

$$\mathrm{a_n = a_0 + n \cdot d}$$

where d is the common difference. For our sequence, a_n = 3 + 3n

Geometric Sequence

In a geometric sequence, each term is obtained by multiplying the previous term by a fixed ratio. For instance, the sequence 2, 4, 8, 16, … has a common ratio of 2. The general formula for a geometric sequence is −

$$\mathrm{a_n = a_0 \cdot r^n}$$

where r is the common ratio. For this example, a_n = 2 ⋅ 2n

Fibonacci Sequence

The Fibonacci sequence is a well-known recursive sequence in which each term is the sum of the previous two. It starts with F₀ = 0 and F₁ = 1, it proceeds as follows:

• F₂ = 1,
• F₃ = 2,
• F₄ = 3, and so forth.

The Fibonacci sequence appears frequently in nature and has applications in fields like computer science and financial modelling.

Other Special Sequences

Let us talk about some other sequences which are quite useful in different scenario.

• Square Numbers: The sequence 1, 4, 9, 16… is the sequence of square numbers. They are defined by a_n = n².
• Triangular Numbers: The triangular number sequence 1, 3, 6, 10… it represents the sum of natural numbers up to n. This is given by −

$$\mathrm{T_n = \frac{n(n + 1)}{2}}$$

These sequences are not only mathematically interesting but also useful in counting and combinatorial problems.

Practical Examples and Applications of Sequences

Let us see some examples to better understand how sequences work and where we can apply them in discrete mathematics.

Example 1: Calculating Terms in a Recursive Sequence

Consider the recursive sequence −

$$\mathrm{a_n = 2a_{n-1} + 3}$$

with a₀ = 1. To calculate the next few terms −

• a₁ = 2 × 1 + 3 = 5
• a₂ = 2 × 5 + 3 = 13
• a₃ = 2 × 13 + 3 = 29

This example shows how each term builds on the previous one. This is creating a sequence that grows quickly.

Example 2: Finding the Closed Formula for an Arithmetic Sequence

Suppose we have an arithmetic sequence starting at 4 with a common difference of 7. The formula for the n-th term is −

$$\mathrm{a_n = 4 + 7n}$$

Thus, a₅, the sixth term, will be 4 + 7 ⋅ 5 = 39.

Example 3: Recognizing a Pattern in Triangular Numbers

Let's take the triangular sequence 1, 3, 6, 10, 15... This adds successive natural numbers. We can derive each term by adding n to the previous term −

• 1 + 2 = 3
• 3 + 3 = 6
• 6 + 4 = 10
• 10 + 5 = 15, and so on.

To find the formula, we use −

$$\mathrm{T_n = \frac{n(n + 1)}{2}}$$

It shows that the 5th term is,

$$\mathrm{T_5 = \frac{5(5 + 1)}{2} = 15}$$

Example 4: Fibonacci Sequence in Nature

The Fibonacci sequence appears in many natural patterns like flowers waves, arrangement of leaves, the branching of trees, and the fruitlets of a pineapple, etc.

If we wanted to find the 7th term in the sequence, we calculate −

• F₀ = 0
• F₁ = 1
• F₂ = 1
• F₃ = 2
• F₄ = 3
• F₅ = 5
• F₆ = 8
• F₇ = 13

So, the 7th Fibonacci number is 13.

Conclusion

In this chapter, we explained the concept of sequences in discrete mathematics. Sequences are ordered lists of numbers or terms following specific rules.

We started with the basics, discussing how sequences can be defined using closed formulas or recursively. We then understood various types of sequences, the arithmetic, geometric, Fibonacci, and some special sequences like square and triangular numbers.

In addition, we provided several practical examples to help you understand the concept in a better way. We also highlighted the different applications and forms sequences can take, from simple patterns to complex recursive definitions.