Math Lesson 12.1.1 - The Definition of Sequences

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Welcome to our Math lesson on The Definition of Sequences, this is the first lesson of our suite of math lessons covering the topic of Working with Term-to-Term Rules in Sequences, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

The Definition of Sequences

In mathematics, an ordered set of objects or numbers, like a₁, a₂, a₃, a₄, a₅, a₆, …, an, are said to be in a sequence, if, as per a certain rule, they have a definite value. Each of the objects or numbers contained in a sequence is called a term.

The difference between a sequence and any other list of items is that in sequences you are able to find the term that comes after the last term shown in the set. For example,

is a sequence, as you can easily find the term that comes after 17. Indeed, it is easy to realize that every successive term is 4 units greater than the previous one, so the next term that comes after 17 is 17 + 4 = 21.

On the other hand, the number list 4, 6, 11, 20, 21, 29, 34, … is not a sequence, as it is impossible to find any common pattern or rule that helps find the term that comes after 34.

As for the number of terms they contain, sequences are classified into two main types: finite and infinite. As the name itself suggests, a finite sequence contains a finite number of terms. For example,

is a finite sequence, as it has a definite first term (6) and also a definite last term (34). Look at the curled brackets that are used to indicate the beginning and the end of the sequence.

On the other hand, the mathematical sentence

is an infinite one, as it is endless on the right. This is indicated by the three dots placed after the fourth term. Likewise, the mathematical sentence

is infinite, as it has no beginning. This fact is indicated by the three dots that precede the first known term -6.

Example 1

Which of the following lists of numbers are mathematical sequences? Write their main features.

1. {0, 3, 7, 12, 18, 25, 33, 41, 50}
2. {-3, 7, 17, 27, …}
3. {…, 4, 13, 20, 25, 28, 29}
4. {0, 8, 11, 17, 29, 34}

Solution 1

1. This is a finite mathematical sequence which has a beginning (0) and an end (50). It is a sequence because the difference between two consecutive terms increases by 1 when moving from left to right. This pattern helps identify the other terms that do not appear in the list.
2. This is an infinite mathematical sequence, as it has a beginning (-3) but not an end. Each successive term is 10 more than the previous one.
3. This is an infinite mathematical sequence, as it has an end (29) but not a beginning. It is a sequence because the difference between two successive terms decreases by 2 when moving from left to right.
4. This is a finite list of numbers but not a sequence, as the terms don't follow a specific pattern.

More Working with Term-to-Term Rules in Sequences Lessons and Learning Resources

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2. Sequences and Series Math tutorial: Working with Term-to-Term Rules in Sequences. Read the Working with Term-to-Term Rules in Sequences math tutorial and build your math knowledge of Sequences and Series
3. Sequences and Series Video tutorial: Working with Term-to-Term Rules in Sequences. Watch or listen to the Working with Term-to-Term Rules in Sequences video tutorial, a useful way to help you revise when travelling to and from school/college
4. Sequences and Series Revision Notes: Working with Term-to-Term Rules in Sequences. Print the notes so you can revise the key points covered in the math tutorial for Working with Term-to-Term Rules in Sequences
5. Sequences and Series Practice Questions: Working with Term-to-Term Rules in Sequences. Test and improve your knowledge of Working with Term-to-Term Rules in Sequences with example questins and answers
6. Check your calculations for Sequences and Series questions with our excellent Sequences and Series calculators which contain full equations and calculations clearly displayed line by line. See the Sequences and Series Calculators by iCalculator™ below.
7. Continuing learning sequences and series - read our next math tutorial: Working with Arithmetic and Geometric Series. How to find the Sum of the First n-Terms of a Series.

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Sequences and Series Calculators by iCalculator™

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