Sequences

In mathematics, a sequence is an ordered list of numbers or other mathematical objects that follow a particular pattern. Sequences are important in many areas of mathematics, including calculus, analysis, number theory, and discrete mathematics.

We come across sequences in many places in real life. For example, the house numbers in a row, salary in successive years (by a fixed amount or a by a fixed percentage), page numbers of a book, etc represent sequences. Let us learn more about sequences along with their types, rules, formulas, and examples.

1.	What is a Sequence?
2.	Order of the Sequence
3.	Finite and Infinite Sequences
4.	Types of Sequences in Math
5.	Series and Partial Sums of Sequences
6.	Rules of Sequences
7.	Sequences Formulas
8.	Finding Missing Numbers in a Sequence
9.	FAQs on Sequences

A sequence is a list of numbers (or elements) that exhibits a particular pattern. Each element in the sequence is called a term. A sequence can be finite, meaning it has a specific number of terms, or infinite, meaning it continues indefinitely. Sequences can be described in different ways, such as an explicit formula, a recurrence relation, or a table of values.

• An explicit formula gives a direct way to compute each term in the sequence.
• A recurrence relation expresses each term in terms of one or more preceding terms.
• A table of values simply lists the terms in the sequence.

There are several types of sequences in math such as arithmetic sequences, quadratic sequences, geometric sequences, triangular sequences, square number sequences, cube number sequences, and triangular number sequences. Let us learn each of the sequences in detail in the upcoming sections.

The order of a sequence can be either ascending or descending order.

Ascending Order

If the elements of the sequence are in increasing order, then the order of the sequence is ascending.

The above sequence is in ascending order as its terms are "increasing" by 2.

Descending Order

If the elements of the sequence are in decreasing order, then the order of the sequence is descending.

The above sequence is in descending order as its terms are "decreasing" by 4.

There are two types of sequences: finite sequences and infinite sequences. It is possible to count the number of terms in a finite sequence. An infinite sequence is a sequence that is not finite.

Finite Sequence

A sequence having a finite number of terms is called a finite sequence. For example, a sequence of the number of bounces a ball takes to come to the rest is a finite sequence.

Infinite Sequence

A sequence having an infinite number of terms is called an infinite sequence. For example, a sequence of natural numbers forms an infinite sequence: 1, 2, 3, 4, and so on.

There are a few special sequences like arithmetic sequence, geometric sequence, Fibonacci sequence, harmonic sequence, triangular number sequence, square number sequence, and cube number sequence. Apart from these, there can be sequences that follow some other pattern. For example, 2, 9, 28, 65, ... is a sequence in which the numbers can be written as 1³ + 1, 2³ + 1, 3³ + 1, 4³ + 1,.... and this sequence does not belong to any of the following sequences.

We will discuss these sequences in detail.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each successive term is a sum of its preceding term and a fixed number. This fixed number is called a common difference. The terms of the arithmetic sequence are of the form a, a+d, a+2d, ....

Example: Mushi put $30 in her piggy bank when she was 7 years old. She increased the amount on her each successive birthday by $3. So, the amount in her piggy bank follows the pattern of $30, $33, $36, and so on. The succeeding terms are obtained by adding a fixed number, that is, $3. This fixed number is called a common difference. It can be positive, negative, or zero.

Quadratic Sequence

We have already seen that if the differences (referred to as first differences) between every two successive terms are the same, then it is called an arithmetic sequence (which is also known as a linear sequence). But if the first differences are NOT the same, and instead, the second differences are the same, then the sequence is known as a quadratic sequence.

Example: The sequence 1, 2, 4, 7, 11, ... is a quadratic sequence because their second differences are the same. Take a look at the figure below.

Geometric Sequence

A geometric sequence is a sequence where every term bears a constant ratio to its preceding term. This ratio is called the "common ratio". The terms of the geometric sequence are of the form a, ar, ar², ....

Example: Consider an example of geometric sequence: 1, 4, 16, 64, .... Here, 4/1 = 16/4 = 64/4 = ... = 4. Hence, it is a geometric sequence with common ratio 4.

Harmonic Sequence

A harmonic sequence is a sequence obtained by taking the reciprocal of the terms of an arithmetic sequence.

Example: We know that the sequence of natural numbers is an arithmetic sequence. So, taking reciprocals of each term, we get 1, 1/2, 1/3, ..., which is a harmonic sequence as their reciprocals 1, 2, 3, ... form an arithmetic sequence.

Triangular Number Sequence

A triangular number sequence is a sequence that is obtained from a pattern forming equilateral triangles. Look at the figure below.

The sequence 1, 3, 6, 10, and so on is a triangular number sequence.

Square Number Sequence

A square number sequence is a sequence that is obtained from a pattern forming squares. Look at the figure below.

The sequence 1, 4, 9, 16, and so on is a square number sequence.

Cube Number Sequence

A cube number sequence is a sequence that is obtained from a pattern forming cubes. Look at the figure below.

The sequence 1, 8, 27, 64, and so on is a cube number sequence.

Fibonacci Sequence

Fibonacci sequence is a sequence where every term is the sum of the last two preceding terms.

Example: A pair of rabbits do not reproduce in their 1st month. Starting from the 2nd month and every subsequent month, they reproduce another pair. Thus, the number of rabbits starting from 1st month are 0, 1, 1, 2, 3, 4, 7, 11, .... This is called the Fibonacci sequence.

Consider a sequence given by a₁, a₂, a₃, a₄, .... Then the sum a₁ + a₂ + a₃ + a₄ + .... is the series associated with the sequence. Series can be represented using sigma notation, ∑. So, the series is represented as ∑_n=1^∞ a_n. The partial sum is a part of the series. The sum up to k terms in the series ∑_n=1^k a_n and it is called the partial sum of the series. Find more differences between a sequence and a series by clicking here.

Example: Consider a sequence of prime numbers: 2, 3, 5, 7, 11, and so on. The series associated with this is ∑_n=1^∞ a_n, where a_n is the n^th prime number. The partial sum up to 4 terms is 2+3+5+7=17.

We can generally have two types of rules for a sequence (if it is geometric/arithmetic).

• Implicit rule where a term is expressed in terms of its previous term.
• Explicit rule where any term can be found using a general formula.

Example: Consider the sequence of odd numbers 3, 5, 7, .... We will define two rules to define n^th term (general term) of this sequence. Note that this sequence is an arithmetic sequence with the first term 3 (a = 3) and the common difference 2 (d = 5 - 3 = 2). Then:

• Implicit rule: a_n = a_n-1 + 2
• Explicit rule: a_n = a + (n - 1) d = 3 + (n - 1) 2 = 3 + 2n - 2 = 2n + 1.

Here, a_n = a + (n - 1) d is the formula for nᵗʰ term of the arithmetic sequence. Let us see more formulas of different types of sequences in the upcoming section.

As we have seen in the previous section, the formula for a sequence is nothing but the formula for its n^th term. Let us see the formulas for n^th term (a_n) of different types of sequences in math.

• Arithmetic sequence: a_n = a + (n - 1) d, where a = the first term and d = common difference.
• Geometric sequence: a_n = ar^n-1, where a = the first term and r = common ratio.
• Fibonacci sequence: a_n+2 = a_n+1 + a_n. The first two terms are 0 and 1.
• Square number sequence: a_n = n².
• Cube number sequence: a_n = n³.
• Triangular number sequence: a_n = ∑_k=1ⁿ n. This can be further evaluated using the sum of natural numbers formula.

Using the above rules/formulas of sequences, we can find the missing numbers of sequences. Sometimes, we don't need to find the general term also to find the missing terms. If the given sequence doesn't belong to any of the specific sequences mentioned above, then we have to observe the pattern of the sequence and define the general term. Using that we can find the missing numbers.

Example: Find the missing number of the sequence 2, 12, 36, 80, __.

Solution:

It is very clear that the sequence does not belong to any of the sequences that we have mentioned in the previous section. So let us observe the terms. We can see that:

• 2 = 1² + 1³
• 12 = 2² + 2³
• 36 = 3² + 3³
• 80 = 4² + 4³

Thus, the missing number would be 5² + 5³ = 25 + 125 = 150.

Important Notes on Sequences:

• In an arithmetic sequence, each successive term is obtained by adding the common difference to its preceding term.
• In a geometric sequence, each successive term is obtained by multiplying the common ratio to its preceding term.
• The reciprocal of terms in harmonic sequence form an arithmetic sequence.

☛ Related Topics:

• Sequence Calculator
• Arithmetic Sequence Calculator
• Geometric Sequence Calculator
• Series Calculator

 

What is the Definition of Sequence?

A sequence is a collection of ordered and indexed items, often numbers arranged according to a certain rule or pattern. In mathematics, sequences are usually represented by a series of terms, each of which corresponds to a particular position or index in the sequence. For example, 3, 7, 11, 15, ... is a sequence as there is a pattern where each term is obtained by adding 4 to its previous term.

What are 4 Types of Sequences?

There are many types of sequences. The following are the 4 important types of sequences:

• Arithmetic sequence
• Geometric sequence
• Harmonic sequence
• Fibonacci sequence

What is the Formula of Sequence?

The general term (or) n^th term defines a sequence. If a sequence belongs to specific types like arithmetic, geometric, etc, then we have formulas to find the general term of the respective sequence.

• For example, the formula for the n^th term of an arithmetic sequence is: a_n = a₁ + (n-1)d, where a₁ is the first term, d is the common difference between terms, and a_n is the nth term.

• Similarly, the formula for the n^th term of a geometric sequence is: a_n = a₁ * r^(n-1), where a₁ is the first term, r is the common ratio between terms, and a_n is the nth term.

What Kind of Sequence is 7, 20, 33, ...?

In the sequence 7, 20, 33, ...,

• second term = 20 = 7 + 13 = first term + 13
• third term = 33 = 20 + 13 = second term + 13

Since every term is obtained by adding 13 to its previous term, it is an arithmetic sequence (which is also known as arithmetic progression).

How to Construct an Arithmetic Sequence?

Follow the steps mentioned below to construct an arithmetic sequence.

• Take any number as the first term.
• Fix a number as a common difference.
• Add the common difference to the first term to obtain the second term.
• To obtain each successive term, keep adding the common difference to its preceding term.

How to Construct a Geometric Sequence?

Follow the steps mentioned below to construct a geometric sequence.

• Take any number as the first term.
• Fix a number as a common ratio.
• Multiply the common ratio to the first term to obtain the second term.
• To obtain each successive term, keep multiplying the common ratio to its preceding term.

What is the Difference Between a Sequence and a Series?

A sequence is a collection of elements that are arranged according to a specific pattern whereas a series is the sum of a few or all elements of the sequence. Here are more differences.

	Sequence	Series
Definition	An ordered list of numbers or other mathematical objects	The sum of the terms in a sequence
Notation	{an} (or) {a1, a2, a3, ...}	Σan (or) a1 + a2 + a3 + ...
Purpose	Describes individual terms in a pattern	Aggregates individual terms into a single value
Calculation	Individual terms can be calculated using the explicit formula, recurrence relation, or other methods	The sum of the terms must be calculated using mathematical operations or techniques such as integration or approximation
Examples	Fibonacci sequence, arithmetic sequence, geometric sequence	Geometric series, telescoping series, power series

How to Find the Sum of Infinite Sequences?

The sum of all infinite sequences may not exist.

• We can find the sum of infinite geometric sequence only when its common ratio (r) is less than 1. If its first term is a, then the sum of its infinite terms is a / (1 - r).
• We can find the sum of an infinite telescoping series by cancelling terms and computing the sum of the rest of the terms.
• To find the sum of a power series, use techniques like differentiation and integration to find a closed-form expression for the sum.