Sequences

What is a Sequence?

Mathematical sequences are a group of terms with a certain rule that dictates a certain operation must be performed and repeated in order to get from one term to the next.
The operation can be addition, subtraction, multiplication, division, or any other mathematical operation.

For example, the following is a basic numerical series:
$1, 2, 3, 4, 5$

To get from one term to the next in the sequence we add $+1$ .
$2 = 1+1$
$3 = 2+1$
$4 = 3+1$
And so on.

Comparison of sequences: The arithmetic sequence starts at -6 and increases by 7 each time (-6, 1, 8, 15, 22). The geometric sequence starts at 1 and multiplies by 3 each time (1, 3, 9, 27, 81). Arrows indicate the operation between term

Detailed explanation

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Test yourself on series / sequences!

12 ☐ 10 ☐ 8 7 6 5 4 3 2 1

Which numbers are missing from the sequence so that the sequence has a term-to-term rule?

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Next, we will present some more complex sequences of numbers, including one of the most famous: the Fibonacci sequence.

The patterns of sequences vary from one to another.

Below, we present some examples of sequences with a different rule for each.

Note that the given rule can appear with: division, multiplication, addition, subtraction, or any combination of these.

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Examples of Sequences

Exercise 1

$2, 4, 8, 16, 32$
In this sequence, to get from one term to the next we will multiply by $2$ .

$2$

$4=2\times2$

$8=2\times4$

$16=2\times8$

And so on.

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Test your knowledge

Question 1

Is there a term-to-term rule for the sequence below?

18 , 22 , 26 , 30

Question 2

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 1,2,3,4,5,6 $$

Question 3

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 10,8,6,4,2 $$

Exercise 2

$3, 9, 27, 81, 243$
In this sequence, to get from one term to the next we need to multiply by $3$ .
$3$

$9=3\times3$

$27=9\times3$

$81=27\times3$

$243=81\times3$

And so on.

Exercise 3

$6, 4, 2, 0, -2$

In this sequence, to get from one term to the next we need to subtract $2$ .

$6$

$4=6-2$

$2=4-2$

$0=2-2$

$-2=0-2$

Do you know what the answer is?

Question 1

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 94,96,98,100,102,104 $$

Question 2

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 13,16,20,23 $$

Question 3

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 13,10,7,4,1 $$

Exercise 4

$1000, 500, 250, 125, 62.5$
In this example, the operation used is division. In order to get from one term to the next, we divide the number by $2$ .

$1000$

$500=1000:2$

$250=500:2$

$125=250:2$

$62.6=125:2$

Exercise 5

$320, 80, 20, 5$

The rule of this sequence is to divide each number by $4$ to find the next number.

$320$

$80=320:4$

$20=80:4$

$5=20:4$

Check your understanding

Question 1

Look at the following set of numbers and determine if there is a rule. If there is one, what is it?

$$ 5,10,15,20,25,30 $$

Question 2

The table shows the number of balls and the number of courts at the school:

Complete:

Number of balls is _________ than the number of courts

Question 3

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 2,4,8,16,32,64 $$

Exercises

Try to work out the rule for each sequence:

$1,3.75,6.5,9.25,12$
$7,49,343,2401,16807$
$0,-15,-30,-45,-60,-75$
$891,297,99,33,11$
$2,8,512,134217728$

Review Questions

What are sequences in mathematics?

Sequences are ordered sets of numbers that follow a rule or pattern.

Do you think you will be able to solve it?

Question 1

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 1,3,9,26,81 $$

Question 2

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 100,50,25,10,20 $$

Question 3

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 256,64,16,4,1 $$

What is a sequence and a sequence rule?

A sequence is a set of ordered numbers. The numbers follow a rule that tells us how to obtain the numbers of the sequence using the previous ones. Many times the rules are governed by the operations of addition, subtraction, multiplication, division, or some combination thereof.

What types of sequences are there in mathematics?

There are many types of sequences. For example, increasing and decreasing sequences, in which the numbers are either increasing or decreasing and following a certain pattern. There are also very famous sequences that have their own name, such as the Fibonacci sequence. In this series, the two previous numbers must be added to obtain the next number.

Test your knowledge

Question 1

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 12,24,35,48,60 $$

Question 2

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 88,66,44,22,2 $$

Question 3

12 ☐ 10 ☐ 8 7 6 5 4 3 2 1

Which numbers are missing from the sequence so that the sequence has a term-to-term rule?

Examples with solutions for Series / Sequences

Exercise #1

12 ☐ 10 ☐ 8 7 6 5 4 3 2 1

Which numbers are missing from the sequence so that the sequence has a term-to-term rule?

Video Solution

Step-by-Step Solution

It is possible to see that there is a difference of one number between each number.

That is, 1 is added to each number and it will be the next number:

$1+1=2$

$2+1=3$

$3+1=4$

Etcetera. Therefore, the next numbers missing in the sequence will be: $8+1=9$

$10+1=11$

Answer

11 , 9

Exercise #2

Is there a term-to-term rule for the sequence below?

18 , 22 , 26 , 30

Video Solution

Step-by-Step Solution

To solve this problem, we'll check the differences between consecutive terms:

The difference between $22$ and $18$ is $22 - 18 = 4$ .
The difference between $26$ and $22$ is $26 - 22 = 4$ .
The difference between $30$ and $26$ is $30 - 26 = 4$ .

All differences between consecutive terms are $4$ , indicating a constant increment. Thus, the sequence is arithmetic with a common difference of $4$ .

The term-to-term rule is: to get the next term, add $4$ to the current term.

Therefore, yes, there is a term-to-term rule for this sequence, given by adding $4$ to the previous term.

Answer

Yes

Exercise #3

Look at the following set of numbers and determine if there is any property, if so, what is it?

$1,2,3,4,5,6$

Video Solution

Step-by-Step Solution

To solve this problem, we need to determine if there's a consistent pattern or rule in the sequence $1, 2, 3, 4, 5, 6$ .

Let's proceed step by step:

Step 1: Analyze the given sequence
The sequence is $1, 2, 3, 4, 5, 6$ .
Step 2: Check for a common difference
Calculate the difference between each consecutive pair of numbers:

$2 - 1 = 1$
$3 - 2 = 1$
$4 - 3 = 1$
$5 - 4 = 1$
$6 - 5 = 1$

From the calculations above, we observe that the difference between each consecutive term is $+1$ .

Conclusion: The sequence is an arithmetic sequence with a common difference of $+1$ .

Therefore, the correct choice is $+1$ .

Answer

$+1$

Exercise #4

Look at the following set of numbers and determine if there is any property, if so, what is it?

$10,8,6,4,2$

Video Solution

Step-by-Step Solution

To solve this problem, we need to analyze whether the set of numbers $10, 8, 6, 4, 2$ has a pattern or property.

Step 1: Observe the difference between consecutive terms:
$8 - 10 = -2$
$6 - 8 = -2$
$4 - 6 = -2$
$2 - 4 = -2$
Step 2: Analyze the result.
We see that the difference between consecutive terms is consistently $-2$ .

This indicates that the terms form an arithmetic sequence with a common difference of $-2$ .

Hence, the property of this set of numbers is that it is an arithmetic sequence with a common difference of $-2$ .

By comparing the possible answer choices, we confirm that the correct choice is number 1: $-2$ .

Answer

$-2$

Exercise #5

Look at the following set of numbers and determine if there is any property, if so, what is it?

$94,96,98,100,102,104$

Video Solution

Step-by-Step Solution

One can observe that the difference between each number is 2.

That is, with each leap the next number increases by 2:

$94+2=96$

$96+2=98$

$98+2=100$

and so forth......

Answer

$+2$

Start practice

What is a Sequence?

Test yourself on series / sequences!

Examples of Sequences

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercises

Review Questions

What are sequences in mathematics?

What is a sequence and a sequence rule?

What types of sequences are there in mathematics?

Examples with solutions for Series / Sequences

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Series / Sequences