Series / Sequences - Examples, Exercises and Solutions

Understanding Series / Sequences

Complete explanation with examples

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.

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

Practice Series / Sequences

Test your knowledge with 39 quizzes

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

\( 256,64,16,4,1 \)

Examples with solutions for Series / Sequences

Step-by-step solutions included
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?

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

Video Solution
Exercise #2

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

18 , 22 , 26 , 30

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

Video Solution
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 1,2,3,4,5,6

Step-by-Step Solution

To solve this problem, we need to determine if there's a consistent pattern or rule in the sequence 1,2,3,4,5,61, 2, 3, 4, 5, 6.

Let's proceed step by step:

  • Step 1: Analyze the given sequence
    The sequence is 1,2,3,4,5,61, 2, 3, 4, 5, 6.
  • Step 2: Check for a common difference
    Calculate the difference between each consecutive pair of numbers:

21=12 - 1 = 1
32=13 - 2 = 1
43=14 - 3 = 1
54=15 - 4 = 1
65=16 - 5 = 1

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

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

Therefore, the correct choice is +1 +1 .

Answer:

+1 +1

Video Solution
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 10,8,6,4,2

Step-by-Step Solution

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

  • Step 1: Observe the difference between consecutive terms:
    810=2 8 - 10 = -2
    68=2 6 - 8 = -2
    46=2 4 - 6 = -2
    24=2 2 - 4 = -2
  • Step 2: Analyze the result.
    We see that the difference between consecutive terms is consistently 2-2.

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

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

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

Answer:

2 -2

Video Solution
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 94,96,98,100,102,104

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

Video Solution

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