Series / Sequences - Examples, Exercises and Solutions

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?

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

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

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

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......

$+2$

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

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

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

What is the first element?

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

$2+4$

$3+7$

$4+10$

$5+13$

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:$1+1$

$1+1$

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

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:

$n^2=8\times8=16$

$64$

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

$2n+2$

Calculate the value of the 11th element.

We calculate by replacing$n=11$

$2\times11+2=$

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

$22+2=24$

$24$

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?

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

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

18 , 22 , 26 , 30

Yes

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

$1,2,3,4,5,6$

$+1$

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

$10,8,6,4,2$

$-2$

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

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

$13,10,7,4,1$

$-3$

Look at the following set of numbers and determine if there is a rule. If there is one, what is it?

$5,10,15,20,25,30$

$+5$

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

$2,4,8,16,32,64$

$\times2$

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

$1,3,9,26,81$

Does not exist