A Sequence Is Defined Recursively Using The Formula $f(n+1) = -0.5 F(n$\]. If The First Term Of The Sequence Is 120, What Is $f(5$\]?A. -15 B. -7.5 C. 7.5 D. 15

A Recursive Sequence: Unraveling the Mystery of $f(n+1) = -0.5 f(n)$

In the realm of mathematics, recursive sequences are a fascinating topic that has captivated the minds of mathematicians and scientists for centuries. A recursive sequence is defined by a formula that uses the previous term to calculate the next term. In this article, we will delve into the world of recursive sequences and explore the formula $f(n+1) = -0.5 f(n)$, where the first term of the sequence is 120. Our goal is to find the value of $f(5)$.

Before we dive into the formula $f(n+1) = -0.5 f(n)$, let's take a step back and understand the concept of recursive sequences. A recursive sequence is a sequence of numbers that is defined by a formula that uses the previous term to calculate the next term. The formula is typically of the form $f(n+1) = g(f(n))$, where $g$ is a function that takes the previous term as input and produces the next term.

The Formula $f(n+1) = -0.5 f(n)$

Now that we have a basic understanding of recursive sequences, let's focus on the formula $f(n+1) = -0.5 f(n)$. This formula states that the next term in the sequence is equal to $-0.5$ times the previous term. In other words, each term in the sequence is obtained by multiplying the previous term by $-0.5$.

Now that we have a clear understanding of the formula $f(n+1) = -0.5 f(n)$, let's calculate the value of $f(5)$. To do this, we need to start with the first term of the sequence, which is 120, and then apply the formula repeatedly to obtain the next terms in the sequence.

Step 1: Calculate $f(2)$

To calculate $f(2)$, we need to apply the formula $f(n+1) = -0.5 f(n)$ to the first term of the sequence, which is 120.

$f(2) = -0.5 f(1) = -0.5 (120) = -60$

Step 2: Calculate $f(3)$

To calculate $f(3)$, we need to apply the formula $f(n+1) = -0.5 f(n)$ to the second term of the sequence, which is $-60$.

$f(3) = -0.5 f(2) = -0.5 (-60) = 30$

Step 3: Calculate $f(4)$

To calculate $f(4)$, we need to apply the formula $f(n+1) = -0.5 f(n)$ to the third term of the sequence, which is 30.

$f(4) = -0.5 f(3) = -0.5 (30) = -15$

Step 4: Calculate $f(5)$

To calculate $f(5)$, we need to apply the formula $f(n+1) = -0.5 f(n)$ to the fourth term of the sequence, which is $-15$.

$f(5) = -0.5 f(4) = -0.5 (-15) = 7.5$

In this article, we explored the formula $f(n+1) = -0.5 f(n)$ and calculated the value of $f(5)$ using the first term of the sequence, which is 120. We applied the formula repeatedly to obtain the next terms in the sequence and finally arrived at the value of $f(5)$, which is 7.5.

The final answer is: C. 7.5
A Recursive Sequence: Unraveling the Mystery of $f(n+1) = -0.5 f(n)$

In the previous article, we explored the formula $f(n+1) = -0.5 f(n)$ and calculated the value of $f(5)$ using the first term of the sequence, which is 120. However, we received many questions from readers who were interested in learning more about recursive sequences and the formula $f(n+1) = -0.5 f(n)$. In this article, we will answer some of the most frequently asked questions about recursive sequences and the formula $f(n+1) = -0.5 f(n)$.

Q: What is a recursive sequence?

A: A recursive sequence is a sequence of numbers that is defined by a formula that uses the previous term to calculate the next term. The formula is typically of the form $f(n+1) = g(f(n))$, where $g$ is a function that takes the previous term as input and produces the next term.

Q: What is the formula $f(n+1) = -0.5 f(n)$?

A: The formula $f(n+1) = -0.5 f(n)$ states that the next term in the sequence is equal to $-0.5$ times the previous term. In other words, each term in the sequence is obtained by multiplying the previous term by $-0.5$.

Q: How do I calculate the value of $f(5)$ using the formula $f(n+1) = -0.5 f(n)$?

A: To calculate the value of $f(5)$ using the formula $f(n+1) = -0.5 f(n)$, you need to start with the first term of the sequence, which is 120, and then apply the formula repeatedly to obtain the next terms in the sequence. The steps are as follows:

• Calculate $f(2)$ by applying the formula to the first term: $f(2) = -0.5 f(1) = -0.5 (120) = -60$
• Calculate $f(3)$ by applying the formula to the second term: $f(3) = -0.5 f(2) = -0.5 (-60) = 30$
• Calculate $f(4)$ by applying the formula to the third term: $f(4) = -0.5 f(3) = -0.5 (30) = -15$
• Calculate $f(5)$ by applying the formula to the fourth term: $f(5) = -0.5 f(4) = -0.5 (-15) = 7.5$

Q: What is the pattern of the sequence $f(n)$?

A: The pattern of the sequence $f(n)$ is obtained by applying the formula $f(n+1) = -0.5 f(n)$ repeatedly. The sequence starts with the first term 120, and each subsequent term is obtained by multiplying the previous term by $-0.5$. The sequence is therefore: 120, -60, 30, -15, 7.5, ...

Q: Can I use the formula $f(n+1) = -0.5 f(n)$ to calculate the value of $f(n)$ for any positive integer $n$?

A: Yes, you can use the formula $f(n+1) = -0.5 f(n)$ to calculate the value of $f(n)$ for any positive integer $n$. However, you need to start with the first term of the sequence, which is 120, and then apply the formula repeatedly to obtain the next terms in the sequence.

Q: What are some real-world applications of recursive sequences?

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

• Finance: Recursive sequences are used to model the growth of investments and the behavior of financial markets.
• Biology: Recursive sequences are used to model the growth of populations and the behavior of ecosystems.
• Computer Science: Recursive sequences are used to model the behavior of algorithms and the growth of data structures.

In this article, we answered some of the most frequently asked questions about recursive sequences and the formula $f(n+1) = -0.5 f(n)$. We hope that this article has provided you with a better understanding of recursive sequences and the formula $f(n+1) = -0.5 f(n)$. If you have any further questions, please don't hesitate to contact us.