Solving Recursive Sequences: Find F(5) With F(1) = 18

Hey there, math enthusiasts! Ever stumbled upon a sequence defined by a tricky recursive formula and wondered how to crack the code? Well, you're in the right place! Today, we're diving deep into a fascinating problem involving a sequence where each term depends on the one before it. Our mission? To find the value of the fifth term, f(5), when we know the first term, f(1), is 18 and the sequence follows the rule f(n+1) = f(n) - 2. Sounds like a puzzle, right? Let's put on our detective hats and solve it together!

Understanding Recursive Sequences

Before we jump into the specifics of our problem, let's take a moment to understand what recursive sequences are all about. In simple terms, a recursive sequence is a sequence where each term is defined based on the previous term or terms. Think of it like a set of dominoes falling – each domino's fall depends on the one before it. The formula that defines this relationship is called a recurrence relation. For instance, in our problem, the recurrence relation is f(n+1) = f(n) - 2. This means that to find any term in the sequence, we need to know the term that comes before it. This might seem a bit circular, but it's a powerful way to define sequences, especially when combined with an initial value or values. This initial value acts as the starting point, the first domino that sets the chain reaction in motion. Without it, we wouldn't know where to begin! Common examples of recursive sequences include the Fibonacci sequence, where each term is the sum of the two preceding ones, and arithmetic sequences, where a constant difference is added to each term to get the next. Understanding the underlying pattern and the initial conditions is key to unraveling any recursive sequence. It's like having the secret code to unlock the sequence's hidden values. So, let's keep this in mind as we tackle our specific problem and see how these concepts come into play.

Breaking Down the Problem: f(n+1) = f(n) - 2

Now, let's zoom in on the specific recursive formula we're dealing with: f(n+1) = f(n) - 2. This formula is the heart of our sequence, dictating how each term is generated from the previous one. It tells us that to get the next term (f(n+1)), we simply subtract 2 from the current term (f(n)). This might seem straightforward, but it's crucial to grasp the implications fully. The constant subtraction of 2 indicates that we're dealing with an arithmetic sequence, where the difference between consecutive terms is always the same. In this case, the common difference is -2. This means the sequence will be decreasing as we move from one term to the next. But where does this sequence start? That's where the initial condition comes in. We're given that f(1) = 18. This is our starting point, the anchor that grounds the entire sequence. It tells us that the first term in the sequence is 18. Now, armed with the recurrence relation and the initial condition, we have all the pieces we need to start calculating subsequent terms. We know how to move from one term to the next, and we know where to begin. It's like having the map and the starting location – now we just need to follow the path to our destination, which in this case is f(5). So, let's roll up our sleeves and start calculating, step by step, how we can use this information to find the value of the fifth term.

Step-by-Step Calculation of f(5)

Okay, let's get our hands dirty and calculate the value of f(5), step by step. We know that f(1) = 18, and we have the formula f(n+1) = f(n) - 2. Our goal is to find f(5), so we need to work our way through the sequence until we reach the fifth term. First, let's find f(2). Using the formula, f(2) = f(1) - 2. Since f(1) is 18, we have f(2) = 18 - 2 = 16. Great! We've found the second term. Now, let's move on to f(3). Again, using the formula, f(3) = f(2) - 2. We just calculated f(2) as 16, so f(3) = 16 - 2 = 14. We're making progress! Next, we need to find f(4). Applying the formula, f(4) = f(3) - 2. We found f(3) to be 14, so f(4) = 14 - 2 = 12. Almost there! Finally, we can calculate f(5). Using the formula one last time, f(5) = f(4) - 2. We calculated f(4) as 12, so f(5) = 12 - 2 = 10. And there you have it! We've successfully navigated the recursive sequence and found that f(5) equals 10. By systematically applying the recurrence relation and using the initial condition, we were able to unravel the sequence and reach our target. This step-by-step approach is a powerful technique for solving recursive sequence problems, allowing us to break down the problem into manageable chunks and avoid getting lost in the process.

The Answer: f(5) = 10

So, after carefully navigating through the recursive sequence, we've arrived at our destination: f(5) = 10. This is the solution to our problem, the value of the fifth term in the sequence. It's a testament to the power of recursive formulas and how they can define sequences in an elegant and concise way. By understanding the recurrence relation and the initial condition, we were able to unlock the hidden value of f(5). But more than just finding the answer, this exercise highlights the beauty of mathematical problem-solving. It's about taking a problem, breaking it down into smaller, manageable steps, and systematically working towards the solution. Each step builds upon the previous one, creating a logical chain that leads us to the final answer. And while the answer itself is important, the process of getting there is just as valuable. It's about developing critical thinking skills, problem-solving strategies, and a deeper understanding of mathematical concepts. So, the next time you encounter a recursive sequence, remember the steps we took today: understand the recurrence relation, identify the initial condition, and calculate step by step until you reach your goal. With practice and perseverance, you'll become a master of unraveling recursive sequences!

Conclusion

In conclusion, we successfully determined that f(5) = 10 for the given recursive sequence f(n+1) = f(n) - 2 with the initial condition f(1) = 18. We explored the concept of recursive sequences, understood the importance of recurrence relations and initial conditions, and applied a step-by-step approach to calculate the desired term. This problem-solving journey not only gave us the answer but also reinforced the valuable skills of critical thinking and systematic problem-solving in mathematics. Remember, tackling recursive sequences is like solving a puzzle – each piece fits together to reveal the final solution. Keep practicing, and you'll become a pro at unraveling these mathematical mysteries! For further exploration of recursive sequences and related topics, you can visit resources like Khan Academy's Sequences and Series section.