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.

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

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

$+2$

Exercise #3

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

Video Solution

Step-by-Step Solution

It is possible to see that if you multiply each number from the right column by 2, you get the number from the left column.

That is: $1\times2=2$

$2\times2=4$

$3\times2=6$

Therefore, the number of balls is 2 times greater than the number of courts.

Answer

2 times greater

Exercise #4

The sequence below is structured according to a term-to-term rule.

What is the first element?

$\text{?}+\text{?}$

$2+4$

$3+7$

$4+10$

$5+13$

Video Solution

Step-by-Step Solution

We start with the right column in the exercises.

Between each number there is a jump of +3: $4+3=7$

$7+3=10$

Etcetera.

Now we move to the left column of the exercises.

Between each number there is a jump of +1:

$2+1=3$

$3+1=4$

Now we can figure out which exercise is missing:

The left digit will be: $2-1=1$

The right digit will be: $4-3=1$

And the missing exercise is: $1+1$

Answer

$1+1$

Exercise #5

Below is a sequence represented by squares. How many squares will there be in the 8th element?

Video Solution

Step-by-Step Solution

It is apparent, that for each successive number, a square is added in length and one in width.

Hence, the rule using the variable n is:

$a(n)=n^2$

Therefore, the eighth term will be:

$n^2=8\times8=16$

Answer

$64$

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