Recurrence Relations

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: $-1$.

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 $-256$.

If you are interested in this article, you may also be interested in the following articles:

• The real line
• Opposite numbers
• Integers
• Absolute value

You will find a variety of useful mathematics articles in Tutorela site!

Is there a valid rule for the following sequence of numbers?

$30,26,22,18$

Solution:

Yes, since to get to the next number we must subtract $4$ from the previous number.

$30-4=26$

$26-4=22$

$22-4=18$

Answer:

Yes, subtract $(4)$.

Describe the recurrence relation using the variable $n$.

$21,24,27,30$

Solution:

To find a formula that describes the recurrence relation we use the formula:

$a_n = a_1+d\left(n-1\right)$

Where $a_1$ corresponds to the first element of the sequence and $d$ to the difference between any two consecutive numbers.

We place the corresponding data in the formula:

$a_n=21+3\left(n-1\right)$

We simplify:

$a_n = 21+3n-3$

$a_n = 3n+18$

Answer:

$a_n = 3n+18$

In the classroom there are $10$ numbered seats.

Complete the seating progression:

$20,18$

$16,14$

__ , __

$8,6$

$2,4$

Solution:

Each time we subtract $4$ from the two sides, therefore:

$14-4=10$

$16-4=12$

Answer:

$12,10$

Describe the recurrence relation using the variable $n$.

$50,75,100$

Solution:

We substitute the data according to the formula:

$a_n=a1+d\left(n-1\right)$

$a_n = 50+25(n-1)$

$a_n = 25n+50-25$

$a_n = 25n+25$

Answer:

$a_n = 25n+25$

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 $7n-2$.

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.

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.

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