Finding F(5) In A Recursive Sequence Defined By F(n+1) = F(n) - 2

Understanding Recursive Sequences

Recursive sequences are a fundamental concept in mathematics, particularly within the study of discrete mathematics and sequences. At its core, a recursive sequence is a sequence where terms are defined using one or more preceding terms. This means that to find a specific term in the sequence, you need to know the values of the terms that come before it. This contrasts with explicit formulas, where you can directly calculate any term in the sequence by plugging in its position number. Recursive formulas are essential tools for describing patterns that evolve step-by-step, mirroring processes found in nature, computer science, and various other fields. The key to working with recursive sequences lies in understanding the relationship between consecutive terms, often expressed as a recursive formula or relation. This formula dictates how each term is generated from its predecessors. Additionally, one or more initial terms, known as base cases, are required to kickstart the sequence. Without these base cases, the recursive definition would be incomplete, leaving the sequence undefined. Recursive sequences are encountered in various mathematical contexts, including arithmetic sequences, geometric sequences, the Fibonacci sequence, and many more complex patterns. Their ability to model step-by-step processes makes them invaluable in fields like computer science (algorithm design), finance (compound interest), and biology (population growth).

In our problem, we are presented with a recursive sequence defined by the formula $f(n+1) = f(n) - 2$. This formula tells us that each term in the sequence is obtained by subtracting 2 from the previous term. We are also given the initial condition $f(1) = 18$, which serves as the starting point for our sequence. Our goal is to find the value of $f(5)$, the fifth term in the sequence. To achieve this, we will iteratively apply the recursive formula, starting from the known value of $f(1)$, until we reach $f(5)$. This step-by-step approach highlights the fundamental nature of recursion, where we build upon previously computed values to determine subsequent ones.

Applying the Recursive Formula Step-by-Step

To find $f(5)$, we will systematically apply the recursive formula $f(n+1) = f(n) - 2$, using the initial condition $f(1) = 18$. We start by finding $f(2)$, then $f(3)$, $f(4)$, and finally $f(5)$. This iterative process demonstrates how each term in the sequence depends on its predecessor, emphasizing the core concept of recursion.

First, let's find $f(2)$. We substitute $n = 1$ into the recursive formula: $f(1+1) = f(1) - 2$. Since we know that $f(1) = 18$, we have $f(2) = 18 - 2 = 16$. So the second term in the sequence is 16.

Next, we find $f(3)$. We substitute $n = 2$ into the recursive formula: $f(2+1) = f(2) - 2$. We just calculated that $f(2) = 16$, so we have $f(3) = 16 - 2 = 14$. Thus, the third term in the sequence is 14.

Now, let's calculate $f(4)$. We substitute $n = 3$ into the recursive formula: $f(3+1) = f(3) - 2$. Since we found that $f(3) = 14$, we have $f(4) = 14 - 2 = 12$. This gives us the fourth term in the sequence, which is 12.

Finally, we can find $f(5)$. We substitute $n = 4$ into the recursive formula: $f(4+1) = f(4) - 2$. We calculated that $f(4) = 12$, so we have $f(5) = 12 - 2 = 10$. Therefore, the fifth term in the sequence is 10. By systematically applying the recursive formula and using the initial condition, we have successfully determined the value of $f(5)$. This step-by-step approach underscores the essence of recursion, where we build upon previously computed terms to find subsequent ones.

The Arithmetic Sequence Pattern

By calculating the first few terms of the sequence, we can observe a distinct pattern. We found that $f(1) = 18$, $f(2) = 16$, $f(3) = 14$, $f(4) = 12$, and $f(5) = 10$. Notice that each term is 2 less than the previous term. This consistent difference between consecutive terms is a hallmark of an arithmetic sequence. An arithmetic sequence is defined as a sequence where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted by d. In our case, the common difference is -2, as we are subtracting 2 from each term to get the next. Recognizing this pattern allows us to understand the underlying structure of the sequence and to predict future terms without having to iteratively apply the recursive formula. For example, we could deduce that $f(6)$ would be 8, and so on.

The general form of an arithmetic sequence is given by $a_n = a_1 + (n-1)d$, where $a_n$ is the nth term, $a_1$ is the first term, and d is the common difference. We can verify that our recursive sequence follows this pattern. In our case, $a_1 = f(1) = 18$ and $d = -2$. Therefore, the nth term of our sequence can be expressed as $f(n) = 18 + (n-1)(-2)$. This explicit formula allows us to directly calculate any term in the sequence without relying on the recursive definition. For instance, to find $f(5)$, we can plug in $n = 5$: $f(5) = 18 + (5-1)(-2) = 18 - 8 = 10$, which matches our previous result obtained through the recursive method. Understanding the arithmetic nature of the sequence provides an alternative way to solve the problem and offers valuable insights into the sequence's behavior. The connection between recursive definitions and explicit formulas is a powerful concept in mathematics, allowing us to represent sequences in different ways and to choose the most efficient method for solving specific problems.

Answer: Therefore, $f(5) = 10$.