Which Is A Recursive Formula For The Sequence $50, 54, 58, 62, \ldots$?A. $a_n = 4n + 46$ B. $\left{\begin{array}{l} A_1 = 50 \ A_n = A_{n-1} + 4 \end{array}\right.$, Where $ N ≥ 2 N \geq 2 N ≥ 2 [/tex] C.
Introduction
Recursive formulas are a fundamental concept in mathematics, used to describe sequences and series. They provide a way to define a sequence recursively, where each term is defined in terms of previous terms. In this article, we will explore the concept of recursive formulas and apply it to a given sequence to find the correct recursive formula.
What is a Recursive Formula?
A recursive formula is a formula that defines a sequence recursively, where each term is defined in terms of previous terms. It is a way to describe a sequence without explicitly listing all the terms. Recursive formulas are often used to solve problems that involve sequences and series.
Example of a Recursive Formula
Let's consider the sequence $50, 54, 58, 62, \ldots$. We can see that each term is obtained by adding 4 to the previous term. This can be represented by the recursive formula:
\left\{\begin{array}{l} a_1 = 50 \\ a_n = a_{n-1} + 4 \end{array}\right.$, where \$n \geq 2
This formula states that the first term is 50, and each subsequent term is obtained by adding 4 to the previous term.
Recursive Formula Options
We are given three options for the recursive formula:
A. $a_n = 4n + 46$
B. $\left{\begin{array}{l} a_1 = 50 \ a_n = a_{n-1} + 4 \end{array}\right.$, where $$
C. (Not provided)
Analyzing Option A
Option A is a formula that defines the nth term of the sequence as $4n + 46$. Let's see if this formula matches the given sequence.
We can plug in the values of n to see if the formula produces the correct terms:
- For n = 1, $a_1 = 4(1) + 46 = 50$
- For n = 2, $a_2 = 4(2) + 46 = 54$
- For n = 3, $a_3 = 4(3) + 46 = 58$
- For n = 4, $a_4 = 4(4) + 46 = 62$
The formula produces the correct terms, but let's see if it matches the recursive formula.
Analyzing Option B
Option B is the recursive formula we derived earlier:
\left\{\begin{array}{l} a_1 = 50 \\ a_n = a_{n-1} + 4 \end{array}\right.$, where \$n \geq 2
This formula states that the first term is 50, and each subsequent term is obtained by adding 4 to the previous term.
We can see that this formula matches the given sequence, and it is a valid recursive formula.
Conclusion
In conclusion, the correct recursive formula for the sequence $50, 54, 58, 62, \ldots$ is:
\left\{\begin{array}{l} a_1 = 50 \\ a_n = a_{n-1} + 4 \end{array}\right.$, where \$n \geq 2
This formula defines the sequence recursively, where each term is obtained by adding 4 to the previous term.
Recursive Formulas in Real-World Applications
Recursive formulas have many real-world applications, including:
- Computer Science: Recursive formulas are used to solve problems in computer science, such as sorting algorithms and graph traversal.
- Economics: Recursive formulas are used to model economic systems and predict economic trends.
- Biology: Recursive formulas are used to model population growth and predict the spread of diseases.
Final Thoughts
Q: What is a recursive formula?
A: A recursive formula is a formula that defines a sequence recursively, where each term is defined in terms of previous terms. It is a way to describe a sequence without explicitly listing all the terms.
Q: How do I know if a formula is recursive?
A: A formula is recursive if it defines each term in terms of previous terms. For example, the formula $a_n = a_{n-1} + 4$ is recursive because it defines each term in terms of the previous term.
Q: What is the difference between a recursive formula and an explicit formula?
A: An explicit formula is a formula that defines each term of a sequence without using previous terms. For example, the formula $a_n = 4n + 46$ is an explicit formula because it defines each term without using previous terms. A recursive formula, on the other hand, defines each term in terms of previous terms.
Q: How do I find the recursive formula for a given sequence?
A: To find the recursive formula for a given sequence, you can try to identify the pattern in the sequence. For example, if the sequence is $50, 54, 58, 62, \ldots$, you can see that each term is obtained by adding 4 to the previous term. This suggests that the recursive formula is $a_n = a_{n-1} + 4$.
Q: Can I use a recursive formula to solve a problem?
A: Yes, recursive formulas can be used to solve a wide range of problems. For example, you can use a recursive formula to model population growth, predict economic trends, or solve optimization problems.
Q: What are some common applications of recursive formulas?
A: Recursive formulas have many real-world applications, including:
- Computer Science: Recursive formulas are used to solve problems in computer science, such as sorting algorithms and graph traversal.
- Economics: Recursive formulas are used to model economic systems and predict economic trends.
- Biology: Recursive formulas are used to model population growth and predict the spread of diseases.
- Finance: Recursive formulas are used to model financial systems and predict stock prices.
Q: How do I write a recursive formula?
A: To write a recursive formula, you need to define each term in terms of previous terms. For example, if you want to define a sequence where each term is obtained by adding 4 to the previous term, you can write the recursive formula as $a_n = a_{n-1} + 4$.
Q: Can I use a recursive formula to solve a problem that has multiple variables?
A: Yes, recursive formulas can be used to solve problems with multiple variables. For example, you can use a recursive formula to model a system with multiple variables, such as a population growth model with multiple species.
Q: What are some common mistakes to avoid when using recursive formulas?
A: Some common mistakes to avoid when using recursive formulas include:
- Infinite recursion: Recursive formulas can lead to infinite recursion if not properly defined.
- Non-termination: Recursive formulas can lead to non-termination if the base case is not properly defined.
- Incorrect initialization: Recursive formulas require proper initialization to ensure correct results.
Q: How do I debug a recursive formula?
A: To debug a recursive formula, you can try the following steps:
- Check the base case: Ensure that the base case is properly defined and initialized.
- Check the recursive case: Ensure that the recursive case is properly defined and leads to a terminating condition.
- Check the initialization: Ensure that the initialization is properly defined and leads to a correct result.
Q: Can I use a recursive formula to solve a problem that has a large number of terms?
A: Yes, recursive formulas can be used to solve problems with a large number of terms. However, you may need to use techniques such as memoization or dynamic programming to optimize the computation and avoid infinite recursion.