### What are Recurrence Relations?

If there is a relationship between the elements of a sequence, the recurrence relation would be the rule that connects them. It is possible to formulate the recurrence relation and use it to find the value of each of the elements of the set according to the position it occupies.

For example

## Ways to Find Recurrence Relations

There are several ways to find recurrence relations. One is to observe the sequence of elements and how they change. Another way is to write down parameters in a table.

A rule can be formulated using addition, subtraction, multiplication or division—or several of these operations together.

Let's look at an example:

Consider the sequence of elements: $3,7,11,15,19$.

If we look closely at the numbers, we can see that there is a certain rule of formation between them: to get from one number to the next, we need to add $4$ each time.

The first element is $3$. If we add $4$ to this number, we will get the second element, which is $7$. If we add $4$ to this number again, we will arrive at the third element ($11$), and so on.

Therefore the rule in this case is: $+4$.

## Examples with solutions for Series

### 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:

$1+1=2$

$2+1=3$

$3+1=4$

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

$10+1=11$

11 , 9

### Exercise #2

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

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

### 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......

$+2$

### 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

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 #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:

$22+2=24$

$24$

### Exercise #6

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

### Step-by-Step Solution

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:

$n^2=8\times8=16$

$64$

### Exercise #7

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 #8

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

18 , 22 , 26 , 30

Yes

### Exercise #9

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 #10

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

$-3$

### Exercise #11

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 #12

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 #13

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 #14

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 #15

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

$88,66,44,22,2$