Series - Examples, Exercises and Solutions

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.

examples with solutions for series

Exercise #1

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

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

Step-by-Step Solution

It can be seen that the difference between each number is 2.

That is, between each jump 2 is added to the next number:

$94+2=96$

$96+2=98$

$98+2=100$

Etcetera

$+2$

Exercise #2

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:

$1+1=2$

$2+1=3$

$3+1=4$

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

$10+1=11$

11 , 9

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

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.

2 times greater

Exercise #4

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

Step-by-Step Solution

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

Therefore, the rule using the variable n is:

$a(n)=n^2$

Therefore, the eighth term will be:

$n^2=8\times8=16$

$64$

Exercise #5

Below is the rule for a sequence written in terms of $n$:

$2n+2$

Calculate the value of the 11th element.

Step-by-Step Solution

We calculate by replacing$n=11$

$2\times11+2=$

First we solve the multiplication exercise and then we add 2:

$22+2=24$

$24$

examples with solutions for series

Exercise #1

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

What is the first element?

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

$2+4$

$3+7$

$4+10$

$5+13$

Step-by-Step Solution

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:$1+1$

$1+1$

Exercise #2

Given a series whose first element is 15, each element of the series is less by 2 of its predecessor.

Is the number 1 an element of the series?

Step-by-Step Solution

We know that the first term of the series is 15.

From here we can easily write the entire series, until we see if we reach 1.

15, 13, 11, 9, 7, 5, 3, 1

The number 1 is indeed an element of the series!

Yes

Exercise #3

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

18 , 22 , 26 , 30

Yes

Exercise #4

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

$+1$

Exercise #5

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

$13,10,7,4,1$

Video Solution

$-3$

examples with solutions for series

Exercise #1

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

$13,16,20,23$

Does not exist

Exercise #2

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

$10,8,6,4,2$

Video Solution

$-2$

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

$+5$

Exercise #4

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

$256,64,16,4,1$

Video Solution

$\times0.25$

Exercise #5

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

$88,66,44,22,2$