Determining the Recursive Formula

Oct 13, 2025 by leolavanderia.com.br 1 views

Introduction

In mathematics, a recursive formula is a way to define a sequence where each term is defined in terms of previous terms. This type of formula is essential in understanding various mathematical concepts, including sequences and series. In this article, we will explore a given sequence and determine which recursive formula can be used to define it for $n > 1$ .

The Given Sequence

The given sequence is $5, 15, 45, 135, 405, 1215, \ldots$ . To determine the recursive formula, we need to analyze the pattern of the sequence. Let's examine the differences between consecutive terms:

$15 - 5 = 10$
$45 - 15 = 30$
$135 - 45 = 90$
$405 - 135 = 270$
$1215 - 405 = 810$

We can observe that the differences between consecutive terms are increasing by a factor of $3$ each time. This suggests that the sequence is obtained by multiplying the previous term by $3$ and then adding a constant.

Analyzing the Sequence

Let's analyze the sequence further by dividing each term by $3$ :

$\frac{15}{3} = 5$
$\frac{45}{3} = 15$
$\frac{135}{3} = 45$
$\frac{405}{3} = 135$
$\frac{1215}{3} = 405$

We can see that the sequence is obtained by multiplying the previous term by $3$ and then adding $0$ . This suggests that the recursive formula is of the form $a_n = 3a_{n-1} + c$ , where $c$ is a constant.

Determining the Recursive Formula

To determine the value of $c$ , we can use the first term of the sequence, which is $5$ . We know that $a_1 = 5$ , and we want to find the value of $a_0$ such that $a_1 = 3a_0 + c$ . Since $a_1 = 5$ , we have:

5 = 3a_0 + c

We also know that $a_0 = 5/3$ , since dividing the first term by $3$ gives us the second term. Substituting this value into the equation above, we get:

5 = 3(5/3) + c

Simplifying the equation, we get:

5 = 5 + c

Subtracting $5$ from both sides, we get:

0 = c

Therefore, the recursive formula is $a_n = 3a_{n-1}$ .

Conclusion

In conclusion, the recursive formula that can be used to define the given sequence for $n > 1$ is $a_n = 3a_{n-1}$ . This formula is obtained by analyzing the pattern of the sequence and determining the value of the constant $c$ .

Answer

The correct answer is A. $a_n = \frac{1}{3} a_{n-1}$ is incorrect, the correct answer is $a_n = 3a_{n-1}$

Introduction

In our previous article, we explored a given sequence and determined which recursive formula can be used to define it for $n > 1$ . In this article, we will provide a Q&A section to further clarify the concepts and provide additional insights.

Q: What is a recursive formula?

A: A recursive formula is a way to define a sequence where each term is defined in terms of previous terms. This type of formula is essential in understanding various mathematical concepts, including sequences and series.

Q: How do I determine the recursive formula for a given sequence?

A: To determine the recursive formula, you need to analyze the pattern of the sequence. Look for the differences between consecutive terms and see if there is a pattern. You can also try dividing each term by a constant to see if the resulting sequence is a known sequence.

Q: What is the difference between a recursive formula and an explicit formula?

A: A recursive formula defines a sequence in terms of previous terms, while an explicit formula defines a sequence in terms of the index $n$ . For example, the recursive formula $a_n = 3a_{n-1}$ defines the sequence in terms of previous terms, while the explicit formula $a_n = 3^n$ defines the sequence in terms of the index $n$ .

Q: Can I use a recursive formula to define a sequence that is not arithmetic or geometric?

A: Yes, you can use a recursive formula to define a sequence that is not arithmetic or geometric. For example, the sequence $a_n = 2a_{n-1} + 1$ is a recursive formula that defines a sequence that is not arithmetic or geometric.

Q: How do I know if a recursive formula is correct?

A: To determine if a recursive formula is correct, you need to check if it satisfies the initial conditions and if it produces the correct sequence. You can also try plugging in different values of $n$ to see if the formula produces the correct terms.

Q: Can I use a recursive formula to solve a problem that involves a sequence?

A: Yes, you can use a recursive formula to solve a problem that involves a sequence. For example, you can use a recursive formula to find the sum of a sequence or to find the value of a term in a sequence.

Q: What are some common mistakes to avoid when working with recursive formulas?

A: Some common mistakes to avoid when working with recursive formulas include:

Not checking if the formula satisfies the initial conditions
Not checking if the formula produces the correct sequence
Not plugging in different values of $n$ to see if the formula produces the correct terms
Not using the correct notation and terminology

Conclusion

In conclusion, recursive formulas are an essential tool in mathematics, and understanding how to use them can help you solve a wide range of problems. By following the tips and guidelines outlined in this article, you can become more confident in your ability to work with recursive formulas and solve problems that involve sequences.

Additional Resources

If you are interested in learning more about recursive formulas and sequences, here are some additional resources that you may find helpful:

Khan Academy: Sequences and Series
MIT OpenCourseWare: Mathematics 18.01: Single Variable Calculus
Wolfram MathWorld: Recursive Formula

Answer

The correct answer is A. $a_n = 3a_{n-1}$