Introduction to Sequences

A sequence is a list of numbers arranged in a specific order, following a particular rule. Each number in the sequence is called a term.

In mathematics, a sequence is an ordered list of numbers, where each number in the list is called a term. Sequences are defined by a specific rule that determines how each term is generated from its predecessor.

Example:

Sequence: 2, 4, 6, 8, 10,…
Rule: Add 2 to the previous term to get the next term.

Order of Sequence

Order of a sequence in mathematics refers to the arrangement of its terms based on their position or value, usually indexed by natural numbers. There are generally two types of order of sequences:

• Ascending Order
• Descending Order

Ascending Order

When we talk about the ascending order of a sequence, we refer to arranging the terms of the sequence from the smallest to the largest. This concept is straightforward and is commonly used to organize data in a manner that is easy to analyze and understand.

Examples of Sequences in Ascending Order:

• Original Sequence: 7, 2, 9, 4, 5
• Ascending Order: 2, 4, 5, 7, 9

• Original Sequence: 2, 5, 8, 11, 14, ...
• Ascending Order: 2, 5, 8, 11, 14, ...

Descending Order

When we talk about the descending order of a sequence, we refer to arranging the terms of the sequence from the largest to the smallest. This is the opposite of ascending order and is commonly used to prioritize data based on magnitude.

Examples of Sequences in Descending Order:

• Original Sequence: 7, 2, 9, 4, 5
• Descending Order: 9, 7, 5, 4, 2

• Original Sequence: 14, 11, 8, 5, 2, ...
• Descending Order: 14, 11, 8, 5, 2, ...

Finite and Infinite Sequences

In mathematics, sequences can be classified as either finite or infinite based on the number of terms they contain. Let's learn about them in detail.

Finite Sequence

A finite sequence has a limited number of terms. It starts at a specific term and ends at a specific term.
Example:

• Sequence of First Five Natural Numbers: 1, 2, 3, 4, 5
• Sequence of First Five Even Numbers: 2, 4, 6, 8, 10

Infinite Sequence

An infinite sequence has an unlimited number of terms. It continues indefinitely without terminating.
Example:

• Sequence of Natural Numbers: 1, 2, 3, 4, 5, ...
• Sequence of Even Numbers: 2, 4, 6, 8, 10, ...

Types of Sequences in Math

There are many types of sequences but mostly four types of sequences are well known, let's take a look at these 4 types,

• Arithmetic Sequences
• Geometric Sequences
• Harmonic Sequences
• Fibonacci Numbers

Arithmetic Sequences

In an arithmetic sequence, the difference between consecutive terms is always the same. For example, the sequence 5, 9, 13, 17... is arithmetic because the difference between consecutive terms is always four.

The difference between a sequence and a progression is that to calculate its  nth term, a progression has a specific formula i.e,

T_n = a + (n - 1)d

Which is the formula of the n^th term of an arithmetic progression.

Geometric Sequences

A geometric sequence goes from one term to the next by always multiplying or dividing by the same value. The number multiplied (or divided) at each stage of a geometrical sequence is named the common ratio.

The formula for the nth term of the geometric sequence is, where a₁ is the first term, r is the common ratio, and an is the n^th term,

a_n = a₁ r^n-1

The sum of n terms in Geometric Sequence is:

Harmonic Sequences

Harmonic sequence, in mathematics, a sequence of numbers a₁, a₂, a₃,… such their reciprocals 1/ a₁, 1/ a₂, 1/ a₃,… form an arithmetic sequence (numbers separated by a common difference). The arithmetic sequence is just the reciprocal of the harmonic sequence.

The nth term for the harmonic sequence where T_n is the nth term, n is the number of terms, and d is a common difference,

T_n~=~\frac{1}{a+(n-1)d} 

Fibonacci Sequences

Fibonacci series 1, 1, 2, 3, 5, 8, ... is an example of a sequence. The Fibonacci sequence is basically a sequence where the next term is the sum of the earlier 2 terms starting with 1.

Other Sequences in Maths are:

Triangular Number Sequence

The triangular number sequence is a pattern of numbers that represent the number of dots that can form an equilateral triangle. The n-th triangular number is the number of dots in a triangle with n dots on each side. Formula for the Triangular Number Sequence is added below:

T_n = n(n + 1)/2

Example: 1, 3, 6, 10, ...

Square Number Sequence

The square number sequence is a pattern of numbers that represent the number of dots that can form a perfect square. Each term in the sequence is the square of an integer.

Formula for Sequence Number Sequence is added below:

S_n = n²

Example: 1, 4, 9, 16, ...

Cube Number Sequence

The cube number sequence is a pattern of numbers that represent the number of smaller cubes that can form a perfect cube. Each term in the sequence is the cube of an integer.

C_n = n³

Example: 1, 8, 27, 64, ...

Rules of Sequences

Sequences follow specific rules that define how each term is generated. These rules can vary widely depending on the type of sequence.

• Difference between consecutive terms is constant in the Arithmetic Sequence.
• Each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio in Geometric Sequence.
• Each term is the sum of the two preceding terms in the Fibonacci Sequence.
• Each term is the square of its position in the sequence in the Square Number Sequence.
• Each term is the cube of its position in the sequence in the Cube Number Sequence.
• Each term is the reciprocal of its position in the sequence in the Harmonic Sequence.

Sequences Formulas

Various sequence formulas are added in the table below:

Read More:

• Arithmetic Progression
• Geometric Progression
• Harmonic Progression

Solved Examples of Sequences

Example 1: Find the 17th term of the following arithmetic progression 6, 10, 14, 18, 22, 26, ...

Solution:

Formula of nth term of an A.P. is T_n = a + (n-1)d

Here, a = 6 and d = (10 - 6) = 4

Therefore, 17th term = 6 + (17 - 1) × 4
= 6 + 16 × 4 = 6 + 64 = 70

Example 2: Find the sum of the following arithmetic progression  2, 7, 12, 17, 22, ...., 52

Solution:

Formula of sum of an A.P. when first and last term is given is: [n / 2](a + l)
Here, a = 2, d = 5 and l = 52
T_n = a + (n - 1)d

Therefore, 52 = 2 + (n - 1 ) × 5
52 = 2 + 5 × n -5 
52 + 3 = 5 × n
55/5 = n
n = 11

Therefore, sum = (11/2) × (2 + 52)
= 11/2 × 54
= 11 × 27 = 297

Example 3: Is the given series a Geometric progression: 2, 4, 8, 32, 64, 128.

Solution:

In a geometric progression, the common ratio is a fixed number but in this series we have two common ratio as 4/2 = 2 and 32/8 = 4

Therefore, it's not a geometrical progression.

Example 4: Find the common ratio of the following series: 3, 6, 12, 24, 48, ...

Solution:

Common Ratio = (Current Term)/ (Preceding Term)

= 12/6 = 2