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.

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.

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.

Observe the following numerical sets and determine if there is a recurrence relation (rule). If so, specify what it is.

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

B. $9,7,3,8,5,0$

C. $9,11,13,15,17$

D. $1,100,98,85,64$

E. $10,9,8,7,6$

Solution:

A. If we look at this sequence, we see that each subsequent number is greater than the one that precedes it by $1.$. Therefore, the recurrence relation is $+1.$.

B. For this sequence, we can see that there is no relationship between its elements. Therefore, there is no recurrence relation here.

C. Looking at this sequence, we can see that each subsequent number is greater than the one preceding it by $2.$. Therefore, the recurrence relation is $+2$.

D. In this sequence, we see that there is no relationship between its elements. Therefore, there is no recurrence relation.

E. If we look at this sequence, we will see that each subsequent number is smaller than the one preceding it by $1.$. Therefore, the rule is $-1$.

Response:

A. There is a recurrence relation: $+1$.

B. No recurrence relation.

C. There is a rule: $+2$.

D. No rule.

E. There is a recurrence relation. It is: $-1$.

Example No. 2

Look at the number groups below and determine if there is a recurrence relation. If there is, specify what it is and work out the next two terms:

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

Solution:

Looking at the sequence, we can see that there is a mixture of positive and negative numbers and it may seem that there is no rule. However, upon closer inspection, we should see that even though there is a combination of positives and negatives, there is still a recurrence relation to the sequence.

If we first ignore the signs, we will see that each subsequent number is equal to twice the previous one. Now, if we return the signs we should discover that each subsequence is created by multiplying the number that precedes it by $-2$.

$2\times-2=-4$

$-4\times-2=8$

$8\times-2=-16$

$-16\times-2=32$

$32\times-2=-64$

Therefore, the rule for this sequence is $\times(-2)$.

Now let's move on to the second part of the exercise and find the next two elements of the sequence.

We will do this by performing exactly the same operation we have just shown:

$-64\times-2=128$

$128\times-2=-256$

Answer:

There is indeed regularity and it is: $\times(-2)$.

The next two elements of the sequence are: $128$ and $-256$.

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A handful of mathematicians decided in advance on a recurrence relation. They found people whose ages matched the rule and placed them in the following progression:

A. $9n+4-2n-2$

B. $x^2+5n-x^2+2n-2$

C. $7n-2$

D. $9n+4-n-6-n$

Figure:

$5+7=12$

$12+7=19$

Is there a $7n$ in the age equation? (The position increases by $1$, so the age increases by $7$)

Let's look at the first element:

$5=7\times n+\text{?}$

We replace $n=1$.

We move $7$ to the corresponding section.

$5-7=?$

$-2=?$

Rule:

$7n-2$

Answer:

B. $x^2+5n-x^2+2n-2=7n-2$

C. $7n-2$

D. $9n+4-n-6-n=7n-2$

Therefore we have 3 correct answers since they are all equal to$7n-2$.

Review Questions

What is a recurrence relation in mathematics?

When we have a set of ordered numbers, we can say that there is a recurrence relation if there is a pattern or rule that connects these numbers.

Do you think you will be able to solve it?

Question 1

Daniel bought a piggy bank. On the first day, he put in $15 and every day he adds $2. Is it possible for Daniel to save exactly $ 29? If so, when?

How do you work out the recurrence relation of a sequence?

To find the recurrence relation we must analyze the set of numbers and try to use the operations of addition, subtraction, multiplication, or division (or a combination thereof) to describe the set.

How do you work out the recurrence relation of geometric figures?

Geometric figures can also have a recurrence relation. To work it out, you must create a numerical sequence that describes them.

Test your knowledge

Question 1

Daniel bought a piggy bank. On the first day, he put in $15 and every day he adds $2. Is it possible for Daniel to save exactly $ 29? If so, when?