Sequence and Series (Introduction)

In mathematics, the word, “sequence” we usually mean that the collection is ordered in such a way that it has an identified first member, second member, third member and so on.
Sequences, following specific patterns are called progressions.

Introduction

Sequence are types of patterns. If the numbers in a list follow a pattern, then we can relate each number in the list to its numerical position with a rule. Such a rule is called sequence. Therefore, a sequence is an ordered list of numbers. Each number in a sequence is a term of the sequence.

Presumably, a sequence continues by following the pattern that the first few “terms” suggest. To understand the pattern, it is also useful to think of a sequence as a function.
Thus,
A real-valued function with domain the natural numbers N = {1, 2, 3, . . .} is a sequence.
Other functions will also be regarded as sequences: the domain might include 0 alongside the positive integers, meaning that the domain is the non-negative integers, \(Z^{ \ge 0}\) = {0, 1, 2, 3, . . .}.
The range of the function is real numbers
in symbols
A sequencee is a function \(f : N \to R\) .

A sequence can be defined with two different ways

1. Recursive definition (Syntactic definition), Implicit
2. Formal definition (Semantic definition), Explicit

Recursive definition

A sequence is described by comparing each term to the one that comes before it, i.e., by defining the later outputs in terms of previous outputs. Such rule of describing sequence is called recursive relation. For example, in a sequence
133,130,127,124,...,
each term after the first term is equal to three less than the previous term.
Therefore, a recursive definition for this sequence contains two parts.

1. an initial condition (the value of the first term): a₁=133
2. a recursive formula (relates each term after the first term to the one before it): a_n=a_n-1-3 for n>1

Example 1: Writing a recursive definition for a sequence

The number of blocks in two dimensional pyramid is a sequence that follows a recursive formula. What is the recursive definition of the sequence?

The solution is as follows.

• Let us count the number of blocks in each pyramid: it is 1,3,6,10,....
• Now, subtract consecutive terms to find out what happens from one term to the next
  a₂-a₁=3-1=2
  a ₃-a₂=6-3=3
  a ₄-a₃=10-6=4
  
• Now, use n to express the relationship between successive terms
  a _n-a_n-1=n
• To write the recursive definition, state the initial condition and the recursive formula
  a ₁=1 and a_n=a_n-1+n

Example 2

What is the 100th term of the pyramid sequence in the example given below?

Solution

To find the explicit formula, expand the first few terms of the pyramid sequence. which is as below.

a1	a2	a3	a4	...	an
1	3	6	10	...	an
1	1+2	1+2+3	1+2+3+4	...	1+=2+3+4+...+n

Therefore,
a _n=1+2+3+4+---+(n-2)+(n-1)+n (1)
Writing in reverse order, we get
a _n=n+(n-1)+(n-2)+...+4+3+2+1 (2)
Adding (1) and (2), we get

an	=1	+2	3	+---+	(n-2)	+(n-1)	+n
an	=n	+(n-1)	+(n-2)	+---+	+3	+2	+1
2an	=(n+1)	+(n+1)	+(n+1)	+---+	+(n+1)	+(n+1)	+(n+1)
2an	=n(n+1)
an	=\(\frac{n(n+1)}{2} \)

Therefore, the explicit formula, for this sequence is
a _n=\( \frac{n(n+1)}{2} \)
Now, we substitute n by 100 to find the 100th term. Which is
a _n=\( \frac{n(n+1)}{2} \)
or a ₁₀₀=\( \frac{100(100+1)}{2}\)
or a ₁₀₀=5050
Now, we define sequence mathematically.

Semantic Definition

A sequence is a function defined on the set \( \mathbb{Z^+}\). For example, f(a_n)=3a_n. There are different types of such sequences. Among them we discuss three basic types of sequence in the following section.

1. Arithmetic sequence
2. Geometric sequence
3. Harmonic sequence