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.
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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 by2.
2
4=2×2
8=2×4
16=2×8
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
Incorrect
Correct Answer:
Yes
Question 2
Look at the following set of numbers and determine if there is any property, if so, what is it?
\( 10,8,6,4,2 \)
Incorrect
Correct Answer:
\( -2 \)
Question 3
Look at the following set of numbers and determine if there is any property, if so, what is it?
\( 1,2,3,4,5,6 \)
Incorrect
Correct Answer:
\( +1 \)
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=3×3
27=9×3
81=27×3
243=81×3
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?
\( 13,10,7,4,1 \)
Incorrect
Correct Answer:
\( -3 \)
Question 2
Look at the following set of numbers and determine if there is any property, if so, what is it?
\( 13,16,20,23 \)
Incorrect
Correct Answer:
Does not exist
Question 3
Look at the following set of numbers and determine if there is any property, if so, what is it?
\( 94,96,98,100,102,104 \)
Incorrect
Correct Answer:
\( +2 \)
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?
\( 5,10,15,20,25,30 \)
Incorrect
Correct Answer:
\( +5 \)
Question 2
Look at the following set of numbers and determine if there is any property, if so, what is it?
\( 100,50,25,10,20 \)
Incorrect
Correct Answer:
Question 3
Look at the following set of numbers and determine if there is any property, if so, what is it?
\( 12,24,35,48,60 \)
Incorrect
Correct Answer:
Does not exist
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?
\( 1,3,9,26,81 \)
Incorrect
Correct Answer:
Does not exist
Question 2
Look at the following set of numbers and determine if there is any property, if so, what is it?
\( 2,4,8,16,32,64 \)
Incorrect
Correct Answer:
\( \times2 \)
Question 3
Look at the following set of numbers and determine if there is any property, if so, what is it?
\( 256,64,16,4,1 \)
Incorrect
Correct Answer:
\( \times0.25 \)
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?
\( 4,8,12,5,20 \)
Incorrect
Correct Answer:
Does not exist
Question 2
Look at the following set of numbers and determine if there is any property, if so, what is it?
\( 88,66,44,22,2 \)
Incorrect
Correct Answer:
Does not exist
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?
Incorrect
Correct Answer:
11 , 9
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
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
Video Solution
Step-by-Step Solution
To solve this problem of finding the rule for the sequence 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:
10−5=5
15−10=5
20−15=5
25−20=5
30−25=5
Step 2: We observe that the difference between each pair of successive numbers is 5, 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.
Answer
+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
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:
16−13=3
20−16=4
23−20=3
Step 2: Analyze the differences.
The differences between consecutive numbers are not consistent: 3,4, and 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 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.