Sequences

Sequences

In this mini-lesson, we will explore the world of sequences in math. You will get to learn about the types of sequences, examples of sequences, formulas for sequences, and other interesting facts around the topic.

You can also check out the playful calculators to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

Let's start by answering a simple question. Have you ever observed patterns in our day-to-day life?

For example, Olivia and Betty are walking along a street to visit their friend, Hailey.

As they walk, they observe a pattern of house numbers on the street.

These numbers are in ascending order.

Olivia and Betty are walking along a street to visit Hailey

Here, these house numbers form a sequence.

Let's go ahead and learn more about sequences. 

Lesson Plan

What Is a Sequence?

Definition of Sequence 

The sequence is a list of numbers (or items) that exhibits a particular pattern.

Example of Sequence

For example, Olivia has been offered a job with a starting monthly salary of $1000 with an annual increment of $500. 

Olivia has been offered a job with salary of $1000.

Can you calculate her monthly salary in the first three years?

It will be $1000, $1500, and $2000

Observe that Olivia's salary over a number of years forms a sequence.


What Is the Order of a Sequence? 

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

Ascending Order

Look at the sequence below.

Sequence in ascending order

Observe that each term is 2 more than the term preceding it.

Descending Order

In the sequence shown below, each term is 4 less than the term preceding it.

Sequence in descending order

Not only this, a sequence can even have an alternate order.

Look at the bracelet shown below.

Sequence in alternate order

Observe that the shapes square, circle, and triangle follow an alternate pattern.


What Are the 4 Types of Sequences?

The four types of sequences in math are given below.

Types of Sequences Examples
Arithmetic Sequence 1, 5, 9, 13, ...
Geometric Sequence 3, 9, 27, 81, ...
Fibonacci Sequence 0, 1, 1, 2, 3, ...
Harmonic Sequence \(\dfrac{1}{5}\), \(\dfrac{1}{10}\), \(\dfrac{1}{15}\), ...

Infinite or Finite Sequences

Finite Sequence

A sequence having a finite number of terms is called a finite sequence.

Example

For example, a sequence of the number of bounces a ball takes to come to the rest.

Infinite Sequence

A sequence having an infinite number of terms is called an infinite sequence.

Example

For example, a sequence of natural numbers forms an infinite sequence: 1, 2, 3, 4, and so on. 


Sequences Like a Set

A sequence is like a set.

The only difference is that a sequence has a specific pattern that exhibits a rule and the same value can be repeated over time.

For example, a sequence containing numbers 0 and 1 can be 1, 1, 0, 0, 1, 1, 0, 0, and so on. 

But a set is just {0, 1}.

Difference between set and sequence

Many Rules of Sequences

Most of the sequences exhibit a rule.

Examples 

  • A sequence of even numbers follows a rule \(2n\), where \(n \in \mathbb{N}\).
  • A sequence of odd numbers follows a rule \(2n+1\), where \(n \in \mathbb{N}\).
  • A sequence of natural numbers follows a rule \(n\), where \(n \in \mathbb{N}\).

Rules of some sequences


Special Sequences in Math

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.

We will discuss these sequences in detail.

Types os sequences in this short lesson


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.

General form of arithmetic sequence

Example

For 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

Mushi loves collecting money on her every birthday

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.


Geometric Sequence

A geometric sequence is a sequence where every term bears a constant ratio to its preceding term.

General form of geometric sequence

Example

Consider an example of geometric sequence: 1, 4, 16, 64, ...

Observe that \(\dfrac{4}{1}=\dfrac{16}{4}=\dfrac{64}{16}=4\)

Illustration of GP

Here, 4 is a common ratio.


Harmonic Sequence

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

The sequence of natural numbers is an arithmetic sequence.

So, taking reciprocals of each term we get:

\[1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}, ...\]

This is harmonic sequence.

Illustration of harmonic sequence


Triangular Number Sequence

A triangular number sequence is a sequence which is obtained from a pattern forming equilateral triangles.

Look at the figure below.

General form of triangular number sequence

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


Square Number Sequence

A square number sequence is a sequence which is obtained from a pattern forming squares.

Look at the figure below.

General form of square number sequence

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


Cube Number Sequence

A cube number sequence is a sequence which is obtained from a pattern forming cubes.

Look at the figure below.

General form of cube number sequence

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

For 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.

General form of Fibonacci sequence

So, the sequence of the number of rabbits is 1, 1, 2, 3, 4, 7, 11, and so on. 


Series and Partial Sums

Consider a sequence given by \(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\), etc.

Then the sum \(a_{1}+a_{2}+a_{3}+a_{4}+...+a_{n}+...\) is the series associated with the sequence.

Series can be represented using sigma notation, \(\sum\).

So, the series is represented as \(\sum_{n=1}^{\infty}a_{n}\).

The partial sum is a part of the series.

The sum up to \(k\) terms in the series \(\sum_{n=1}^{\infty}a_{n}\) given by \(\sum_{n=1}^{k}a_{n}\) is partial sum of series.

Example

Consider a sequence of prime numbers: 2, 3, 5, 7, 11, and so on. 

The series associated with this is \(\sum_{n=1}^{\infty}a_{n}\), where \(a_{n}\) is the \(n^{\text{th}}\) prime number.

Partial sum up to 4 terms is \(2+3+5+7=17\)


Finding a Rule

We can have more than one rule for a sequence.

For example, the sequence of odd numbers: \(3, 5, 7, 9, ...\) 

We can have many rules to define this sequence.

  • \(2n+1\), where \(n \in \mathbb{N}\)
  • \(a_{1}=3, a_{n+1}=a_{n}+2\), where \(n \in \mathbb{N}\)

We can infact find more rules defining this sequence.

So, now you know the difference between "A Rule" and "The Rule"!


What Is the Formula for Sequences?

Let's learn the formulas for some special sequences.

Arithmetic Sequence

Let the first term of the sequence be \(a\) and the common difference be \(d\).

Then, \(n^{\text{th}}\) term of the sequence is given by: 

\[\begin{align}a_{n}=a+(n-1)d\end{align}\]

Geometric Sequence

Let the first term of the sequence be \(a\) and the common ratio be \(r\).

Then, \(n^{\text{th}}\) term of the sequence is given by:

\[\begin{align}a_{n}=ar^{n-1}\end{align}\]

Harmonic Sequence

Let the first term and the common difference of the associated arithmetic sequence be \(a\) and \(d\) respectively.

Then, \(n^{\text{th}}\) term of the sequence is given by:

\[\begin{align}a_{n}=\dfrac{1}{a+(n-1)d}\end{align}\]

Fibonacci Sequence

Let first two terms of the sequence be \(a_{1}=0\) and \(a_{2}=1\).

Then, \(n^{\text{th}}\) term of the sequence is given by:

\[\begin{align}a_{n+2}=a_{n}+a_{n+1}\end{align}\]

Square Number Sequence

The \(n^{\text{th}}\) term of the sequence is given by:

\[\begin{align}a_{n}=n^{2}\end{align}\]

Triangular Number Sequence

The \(n^{\text{th}}\) term of the sequence is given by the sum of first \(n\) natural numbers.

\[\begin{align}a_{n}=\sum_{k=1}^{n}k\end{align}\]

Cube Number Sequence

The \(n^{\text{th}}\) term of the sequence is given by:

\[\begin{align}a_{n}=n^{3}\end{align}\]

Calculator of Sequences

Try the simulation below to \(n^{\text{th}}\) term of these special sequences.

Hope this calculator of sequences helped you to practice well.


Finding Missing Numbers

Use the formulas of special sequences to find the missing terms.

Examples

For example, consider arithmetic sequence: -32, -29, -26, etc.

The 11th term will be given by \(a+10d\).

Here, \(a=-32\) and \(d=-29+32=3\).

So, \[11^{\text{th}}\text{term}=a+10d=-32+10(3)=-2\]

Experiment with the simulation below to find the missing terms of different sequences.


Finding a Difference 

In an arithmetic sequence, successive terms are obtained by adding a fixed to its preceding terms.

This fixed number is a common difference.

Let \(a\) be the first term and \(d\) be a common difference.

Then, arithmetic sequence is \(a, a+d, a+2d, etc.\).

Common difference in AP

Example 

Consider the sequence: \(2, 5, 8, 11, etc.\)

Common difference is \(5-2=3\)

The number 3 is added every time to obtain the next term.

Now, consider the sequence: \(-3, -7, -11, -15, etc.\)

Common difference is \(-7-(-3)=-7+3=-4\)


Second Differences

Consider a sequence: \(1, 2, 4, 7, 11, etc.\)

The first difference is the difference between consecutive terms.

The second difference is the difference between consecutive first differences.

Second difference in sequences

 
important notes to remember
Important Notes
  1. In an arithmetic sequence, each successive term is obtained by adding the common difference to its preceding term.

  2. In a geometric sequence, each successive term is obtained by multiplying the common ratio to its preceding term.

  3. The reciprocal of terms in harmonic sequence form an arithmetic sequence.

Solved Examples on Sequences

Example 1

 

 

A taxi charges $2 for the first mile and $1.5 for each subsequent mile.

Katie will pay taxi fare to driver.

How much Katie need to pay to the taxi driver if she travels 5 miles?

Solution

The taxi fare for the first few miles are $2, $3.5, $5, ...

It forms an arithmetic sequence with a common difference, \(d=$1.5\) and first term, \(a=$2\)

The 5th term of the sequence will give the taxi fare for the first 5 miles.

The 5th term of the sequence will be given by \(a+4d\).

\[\begin{align}a+4d&=$2+4($1.5)\\&=$2+$6\\&=$8\end{align}\]

\(\therefore\) So, Katie pays \($8\) to the taxi driver for 5 miles.
Example 2

 

 

Flora loves growing flowers in her garden.

There is 1 flower in the 1st row, 4 flowers in the 2nd row, 9 flowers in the 3rd row, and so on.

Flora loves growing flowers in her garden.

In which row did she plant 100 flowers?

Solution

Observe that the number of flowers follows a square number sequence.

\(n^{\text{th}}\) term in the square sequence is given by \(n^2\).

So, in the 10th row, she plants 100 flowers.

\(\therefore\) Flora planted 100 flowers in the 10th row.
Example 3

 

 

Look at the pattern shown below.

Pattern showing an example of Geometric sequence

Observe that each square is half of the size of the square next to it.

Which sequence does this pattern represent?

Solution

Let's write the sequence represented in the figure.

\[1, \dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{8}, \dfrac{1}{16},...\]

Every successive term is obtained by dividing its preceding term by 2

The sequence exhibits a common ratio of \(\dfrac{1}{2}\)

\(\therefore\) The pattern represents the geometric sequence.

Example 4

 

 

Hailey's teacher asks her to fill the missing numbers in the sequences shown on the blackboard.

Examples of Fibonacci and Geometric sequence

Can you help her?

Solution

1. The first sequence is a Fibonacci sequence.

Every term is the sum of the last two terms.

4th term is missing in the sequence.

So, the 4th term is the sum of the 2nd term and 3rd term.

\[\begin{align}4^{\text{th}}\text{term}&=2^{\text{nd}}\text{term}+3^{\text{rd}}\text{term}\\&=8+13\\&=21\end{align}\]

2. The second sequence is a geometric sequence.

Every term is the product of the common ratio and the previous term

4th term is missing in the sequence.

The common ratio is \(\dfrac{15}{3}=5\)

So, the 4th term is the product of 5 and the 3rd term.

\[\begin{align}4^{\text{th}}\text{term}&=\text{Common ratio} \times 3^{\text{rd}}\text{term}\\&=5 \times 75\\&=375\end{align}\]

\(\therefore\) The missing terms are 21 and 375 respectively.
Example 5

 

 

The 6th term and 11th term of the harmonic sequence are 10 and 18 respectively.

Can you find the common difference between the associated arithmetic sequence?

Solution

By definition of the harmonic sequence, we have

6th term of arithmetic sequence = \(\dfrac{1}{10}\)

11th term of arithmetic sequence = \(\dfrac{1}{18}\)

By definition of an arithmetic sequence, we have 2 equations.

\[\begin{align}a+5d=\dfrac{1}{10}\;\;\;\;\;\; \cdots (1)\end{align}\]

\[\begin{align}a+10d=\dfrac{1}{18}\;\;\;\;\;\; \cdots (2)\end{align}\]

Subtract equation (1) from equation (2)

\[\begin{align}(a+10d)-(a+5d)&=\dfrac{1}{18}-\dfrac{1}{10}\\5d&=-\dfrac{2}{45}\\d&=-\dfrac{2}{225}\end{align}\]

\(\therefore\) So, the common difference is \(-\dfrac{2}{225}\)
 
Thinking out of the box
Think Tank
1. The summer Olympics are held after every 4 years.
 

Summer Olympics are held after every 4 years

  Can you think of any formula to represent the years of the summer Olympics starting from 2012?


Interactive Questions

Here are a few activities for you to practice.

Select/type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Important Topics


Let's Summarize

We hope you enjoyed learning about sequences with the examples and practice questions. Now, you will be able to easily remember the formulas of sequence and solve problems on sequences in math, which include arithmetic sequence, geometric sequence, harmonic sequence, and other types of sequences.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions (FAQs)

1. What is the process of constructing arithmetic and geometric 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.

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.
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus