Given The Recursive Formula, Find The First Four Terms:$\[ \begin{align*} a_n &= A_{n-1} + 5 \\ a_1 &= -16 \end{align*} \\]Options:A. -16, -21, -26, -31B. -16, -80, -500, -200C. -16, -11, -6, -1D. 5, -11, -27, -43

Understanding Recursive Formulas

A recursive formula is a way of defining a sequence where each term is defined in terms of the previous term. In this case, we are given a recursive formula for a sequence $a_n$, where each term is defined as the sum of the previous term and a constant value of 5.

The Recursive Formula

The recursive formula is given by:

$$a_n = a_{n-1} + 5$$

where $a_1 = -16$.

Finding the First Four Terms

To find the first four terms of the sequence, we can use the recursive formula to calculate each term in turn.

Calculating the Second Term

The second term, $a_2$, is calculated by adding 5 to the first term, $a_1$:

$$a_2 = a_1 + 5 = -16 + 5 = -11$$

Calculating the Third Term

The third term, $a_3$, is calculated by adding 5 to the second term, $a_2$:

$$a_3 = a_2 + 5 = -11 + 5 = -6$$

Calculating the Fourth Term

The fourth term, $a_4$, is calculated by adding 5 to the third term, $a_3$:

$$a_4 = a_3 + 5 = -6 + 5 = -1$$

The First Four Terms

The first four terms of the sequence are:

• $a_1 = -16$
• $a_2 = -11$
• $a_3 = -6$
• $a_4 = -1$

Conclusion

In this article, we have used a recursive formula to find the first four terms of a sequence. We have shown how to calculate each term in turn using the recursive formula, and have found the first four terms of the sequence to be $-16$, $-11$, $-6$, and $-1$.

Answer

The correct answer is:

• C. -16, -11, -6, -1

Discussion

Q: What is a recursive sequence?

A: A recursive sequence is a sequence where each term is defined in terms of the previous term. In other words, each term is calculated using the previous term and a set of rules.

Q: How do I know if a sequence is recursive?

A: A sequence is recursive if it can be defined using a recursive formula, which is a formula that defines each term in terms of the previous term. For example, the sequence $a_n = a_{n-1} + 5$ is recursive because each term is defined in terms of the previous term.

Q: What is the difference between a recursive sequence and an iterative sequence?

A: An iterative sequence is a sequence where each term is calculated using a set of rules, but the rules do not depend on the previous term. In other words, each term is calculated independently of the previous term. For example, the sequence $a_n = 2n$ is iterative because each term is calculated using a simple formula that does not depend on the previous term.

Q: How do I find the first few terms of a recursive sequence?

A: To find the first few terms of a recursive sequence, you can use the recursive formula to calculate each term in turn. For example, if the recursive formula is $a_n = a_{n-1} + 5$, you can calculate the first few terms as follows:

• $a_1 = -16$
• $a_2 = a_1 + 5 = -16 + 5 = -11$
• $a_3 = a_2 + 5 = -11 + 5 = -6$
• $a_4 = a_3 + 5 = -6 + 5 = -1$

Q: How do I know if a recursive sequence is convergent or divergent?

A: A recursive sequence is convergent if the terms of the sequence approach a limit as $n$ approaches infinity. In other words, the sequence converges to a fixed value. A recursive sequence is divergent if the terms of the sequence do not approach a limit as $n$ approaches infinity. In other words, the sequence diverges to infinity.

Q: What are some common types of recursive sequences?

A: Some common types of recursive sequences include:

• Arithmetic sequences: These are sequences where each term is the previous term plus a fixed constant. For example, the sequence $a_n = a_{n-1} + 5$ is an arithmetic sequence.
• Geometric sequences: These are sequences where each term is the previous term multiplied by a fixed constant. For example, the sequence $a_n = a_{n-1} \cdot 2$ is a geometric sequence.
• Fibonacci sequences: These are sequences where each term is the sum of the previous two terms. For example, the sequence $a_n = a_{n-1} + a_{n-2}$ is a Fibonacci sequence.

Q: How do I use recursive sequences in real-world applications?

A: Recursive sequences have many real-world applications, including:

• Computer science: Recursive sequences are used in algorithms for sorting and searching data.
• Finance: Recursive sequences are used in models for predicting stock prices and interest rates.
• Biology: Recursive sequences are used in models for predicting population growth and disease spread.

Conclusion

In this article, we have answered some common questions about recursive sequences, including what they are, how to find the first few terms, and how to determine if a sequence is convergent or divergent. We have also discussed some common types of recursive sequences and their real-world applications.