Recurrence Relations Practice Problems and Solutions

Understanding Series

Complete explanation with examples

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: $+4$ .

Detailed explanation

Practice Series

Test your knowledge with 40 quizzes

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

$$ 12,24,35,48,60 $$

Examples with solutions for Series

Step-by-step solutions included

Exercise #1

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

$13,16,20,23$

Step-by-Step Solution

To solve this problem, we'll check for consistent differences between the numbers, as this can indicate a property such as an arithmetic sequence.

Step 1: Calculate the difference between each pair of consecutive numbers.

Let's look at the differences:

$16 - 13 = 3$

$20 - 16 = 4$

$23 - 20 = 3$

Step 2: Analyze the differences.

The differences between consecutive numbers are not consistent: $3, 4,$ and $3$ .

This irregularity shows that there is no single property like a consistent common difference, which would indicate an arithmetic sequence.

Therefore, no particular property applies to this set as a whole based on the differences analyzed.

The correct choice is that a regular property does not exist among these numbers.

Therefore, the solution to the problem is: Does not exist.

Answer:

Does not exist

Video Solution

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......

Answer:

$+2$

Video Solution

Exercise #3

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

$10,8,6,4,2$

Step-by-Step Solution

To solve this problem, we need to analyze whether the set of numbers $10, 8, 6, 4, 2$ has a pattern or property.

Step 1: Observe the difference between consecutive terms:
$8 - 10 = -2$
$6 - 8 = -2$
$4 - 6 = -2$
$2 - 4 = -2$
Step 2: Analyze the result.
We see that the difference between consecutive terms is consistently $-2$ .

This indicates that the terms form an arithmetic sequence with a common difference of $-2$ .

Hence, the property of this set of numbers is that it is an arithmetic sequence with a common difference of $-2$ .

By comparing the possible answer choices, we confirm that the correct choice is number 1: $-2$ .

Answer:

$-2$

Video Solution

Exercise #4

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

$13,10,7,4,1$

Step-by-Step Solution

To solve this problem, we will determine if the given sequence of numbers follows a particular pattern or property:

First, we list the sequence provided: $13, 10, 7, 4, 1$ .

Since an arithmetic sequence is one of the simplest patterns, we will check for a common difference, which involves subtracting each term from the next:

Calculate the difference between the first and second terms: $10 - 13 = -3$ .
Calculate the difference between the second and third terms: $7 - 10 = -3$ .
Calculate the difference between the third and fourth terms: $4 - 7 = -3$ .
Calculate the difference between the fourth and fifth terms: $1 - 4 = -3$ .

We observe that the difference between each consecutive pair of numbers is consistently $-3$ . This implies that the sequence has a common difference of $-3$ , and therefore, it is an arithmetic sequence.

In conclusion, the identified property for the sequence is that it is an arithmetic sequence with a common difference of $-3$ .

Therefore, the solution to the problem is $-3$ .

Answer:

$-3$

Video Solution

Exercise #5

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: $8+1=9$

$10+1=11$

Answer:

11 , 9

Video Solution

Frequently Asked Questions

Everything you need to know about Series

What is a recurrence relation and how do I find it?

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A recurrence relation is a rule that connects consecutive terms in a sequence. To find it, look for patterns by examining the difference, ratio, or operation between consecutive terms. For example, in the sequence 3, 7, 11, 15, each term increases by 4, so the recurrence relation is +4.

How do you identify if a sequence has a recurrence relation?

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Check if there's a consistent mathematical operation connecting consecutive terms. Look for: 1) Constant differences (arithmetic sequences), 2) Constant ratios (geometric sequences), 3) Consistent addition, subtraction, multiplication, or division patterns. If no pattern exists, there's no recurrence relation.

What's the formula for arithmetic sequence recurrence relations?

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For arithmetic sequences, use the formula a_n = a_1 + d(n-1), where a_1 is the first term, d is the common difference, and n is the position. For example, if the sequence is 21, 24, 27, 30, then a_n = 21 + 3(n-1) = 3n + 18.

Can sequences with negative numbers have recurrence relations?

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Yes, sequences with negative numbers can have recurrence relations. For example, in 2, -4, 8, -16, 32, -64, each term is multiplied by -2 to get the next term. The pattern involves both sign changes and multiplication, creating the rule ×(-2).

What are common mistakes when finding recurrence relations?

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Common mistakes include: 1) Not checking all consecutive pairs for consistency, 2) Ignoring sign patterns in sequences with negative numbers, 3) Assuming a pattern exists when there isn't one, 4) Mixing up arithmetic and geometric progression formulas, 5) Not simplifying the final formula properly.

How do you find the next terms in a sequence using recurrence relations?

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Once you identify the recurrence relation, apply the same rule to find subsequent terms. For instance, if the rule is +4 and the last term is 18, the next term is 18 + 4 = 22. For multiplication rules like ×(-2), multiply the last term by -2 to get the next term.

What's the difference between arithmetic and geometric recurrence relations?

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Arithmetic sequences have constant differences between consecutive terms (like +3 or -5), while geometric sequences have constant ratios (like ×2 or ×(-1/2)). Arithmetic uses addition/subtraction, geometric uses multiplication/division. Their formulas are different: a_n = a_1 + d(n-1) for arithmetic, a_n = a_1 × r^(n-1) for geometric.

Can you have recurrence relations with mixed operations?

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Yes, some sequences can involve combinations of operations, but they must follow a consistent pattern. However, the most common and easily identifiable recurrence relations involve single operations like constant addition, subtraction, multiplication, or division applied consistently throughout the sequence.

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Recurrence Relations Practice Problems and Solutions

Understanding Series

What are Recurrence Relations?

Ways to Find Recurrence Relations

Practice Series

Examples with solutions for Series

Frequently Asked Questions

What is a recurrence relation and how do I find it?

How do you identify if a sequence has a recurrence relation?

What's the formula for arithmetic sequence recurrence relations?

Can sequences with negative numbers have recurrence relations?

What are common mistakes when finding recurrence relations?

How do you find the next terms in a sequence using recurrence relations?

What's the difference between arithmetic and geometric recurrence relations?

Can you have recurrence relations with mixed operations?

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