Sequences

🏆Practice series / 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 1, 2, 3, 4, 5

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


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Test yourself on series / sequences!

einstein

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.

some more complex sequences of numbers


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

2 2

4=2×2 4=2\times2

8=2×4 8=2\times4

16=2×8 16=2\times8

And so on.


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Exercise 2

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

9=3×3 9=3\times3

27=9×3 27=9\times3

81=27×3 81=27\times3

243=81×3 243=81\times3

And so on.


Exercise 3

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

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

6 6

4=62 4=6-2

2=42 2=4-2

0=22 0=2-2

2=02 -2=0-2


Do you know what the answer is?

Exercise 4

1000,500,250,125,62.5 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 2 .

1000 1000

500=1000:2 500=1000:2

250=500:2 250=500:2

125=250:2 125=250:2

62.6=125:2 62.6=125:2


Exercise 5

320,80,20,5 320, 80, 20, 5

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

320 320

80=320:4 80=320:4

20=80:420=80:4

5=20:4 5=20:4


Check your understanding

Exercises

Try to work out the rule for each sequence:

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

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

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 1+1=2

2+1=3 2+1=3

3+1=4 3+1=4

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

10+1=11 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 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 94+2=96

96+2=98 96+2=98

98+2=100 98+2=100

and so forth......

Answer

+2 +2

Exercise #3

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 5,10,15,20,25,30

Video Solution

Step-by-Step Solution

To solve this problem of finding the rule for the sequence 5,10,15,20,25,30 5, 10, 15, 20, 25, 30 , we will follow these steps:

  • Step 1: Analyze the difference between consecutive numbers in the sequence.
  • Step 2: Identify a consistent pattern or rule.
  • Step 3: Compare the pattern against the given multiple-choice answers.

Now, let's work through each step:

Step 1: Calculate the difference between consecutive terms:

105=510 - 5 = 5

1510=515 - 10 = 5

2015=520 - 15 = 5

2520=525 - 20 = 5

3025=530 - 25 = 5

Step 2: We observe that the difference between each pair of successive numbers is 55, which is consistent throughout the sequence.

Step 3: Compare this pattern with the given choices. The choice corresponding to adding 5 consistently matches our observed pattern.

Therefore, the rule for this sequence is to add 5 to each preceding number to obtain the next number in the sequence. This corresponds with choice number 2: +5 +5 .

Answer

+5 +5

Exercise #4

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

13,16,20,23 13,16,20,23

Video Solution

Step-by-Step Solution

To solve this problem, we'll check for consistent differences between the numbers, as this can indicate a property such as an arithmetic sequence.

  • Step 1: Calculate the difference between each pair of consecutive numbers.

Let's look at the differences:

1613=3 16 - 13 = 3

2016=4 20 - 16 = 4

2320=3 23 - 20 = 3

  • Step 2: Analyze the differences.

The differences between consecutive numbers are not consistent: 3,4, 3, 4, and 3 3 .

This irregularity shows that there is no single property like a consistent common difference, which would indicate an arithmetic sequence.

Therefore, no particular property applies to this set as a whole based on the differences analyzed.

The correct choice is that a regular property does not exist among these numbers.

Therefore, the solution to the problem is: Does not exist.

Answer

Does not exist

Exercise #5

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 2222 and 1818 is 2218=422 - 18 = 4.
  • The difference between 2626 and 2222 is 2622=426 - 22 = 4.
  • The difference between 3030 and 2626 is 3026=430 - 26 = 4.

All differences between consecutive terms are 44, indicating a constant increment. Thus, the sequence is arithmetic with a common difference of 44.

The term-to-term rule is: to get the next term, add 44 to the current term.

Therefore, yes, there is a term-to-term rule for this sequence, given by adding 44 to the previous term.

Answer

Yes

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