Unlocking Sequences How To Find The Sequence Of F(n+1) = F(n) - 2

by Axel Sørensen 66 views

Hey there, math enthusiasts! Today, we're diving into the fascinating world of sequences and functions. We're going to explore a specific function and figure out the sequence it generates. So, buckle up and let's get started!

Understanding the Function

Before we jump into the sequence, let's make sure we understand the function we're working with. The function is defined as f(n+1) = f(n) - 2, with the initial condition f(1) = 10. This is a recursive function, meaning that each term in the sequence is defined in terms of the previous term. In simpler words, to find the next number in the sequence, we need to know the current number. The f(1) = 10 part tells us that the first term in the sequence is 10. Now, the f(n+1) = f(n) - 2 part is the heart of the function. It tells us how to get from one term to the next. Specifically, it says that to find the (n+1)-th term, we take the n-th term and subtract 2 from it. This "subtract 2" part is crucial; it's what gives the sequence its pattern. So, we start with 10, and each subsequent term is 2 less than the previous one. This type of function, where you add or subtract a constant value to get the next term, is called an arithmetic sequence. Arithmetic sequences have a constant difference between consecutive terms, which in our case is -2. This constant difference is also known as the common difference. Understanding this basic structure is key to predicting the sequence's behavior and finding any term we want without having to calculate all the preceding ones. We can already start to imagine what the sequence will look like: it will be a series of numbers decreasing by 2 each time. Let's put this into practice and start generating the sequence.

Generating the Sequence

Now that we understand the function, let's generate the sequence! We know that f(1) = 10, so the first term is 10. To find the second term, f(2), we use the function: f(2) = f(1) - 2 = 10 - 2 = 8. So, the second term is 8. Let's keep going! To find the third term, f(3), we use the function again: f(3) = f(2) - 2 = 8 - 2 = 6. The third term is 6. Notice the pattern? We're subtracting 2 from each term to get the next one. Let's find a few more terms to make sure we've got it. For the fourth term, f(4): f(4) = f(3) - 2 = 6 - 2 = 4. And for the fifth term, f(5): f(5) = f(4) - 2 = 4 - 2 = 2. So, the first five terms of the sequence are 10, 8, 6, 4, and 2. We can continue this process to find as many terms as we want. The sequence will keep decreasing by 2 indefinitely. It's like counting backwards by 2s, starting from 10. This straightforward process highlights the beauty of recursive functions: they define a sequence in a simple, iterative way. Each term builds upon the previous one, creating a clear and predictable pattern. By repeatedly applying the function's rule, we can unravel the sequence and understand its behavior. Now that we've generated the first few terms, let's compare our sequence with the options given and see which one matches.

Identifying the Correct Sequence

Okay, so we've generated the first few terms of the sequence: 10, 8, 6, 4, 2. Now, let's take a look at the options provided and see which one matches our sequence.

  • A. -10, -12, -14, -16, -18, ... This sequence starts with -10 and decreases by 2 each time. This is not the sequence we generated. Our sequence starts at 10 and decreases by 2.
  • B. -2, 8, 18, 28, 38, ... This sequence is increasing, which is the opposite of what our function does. So, this is definitely not the correct answer.
  • C. 8, 18, 28, 38, 48, ... Similar to option B, this sequence is also increasing. Our sequence should be decreasing, so this option is incorrect.
  • D. 10, 8, 6, 4, 2, ... This sequence starts with 10 and decreases by 2 each time, which perfectly matches the sequence we generated. This looks like our winner!

So, after comparing our generated sequence with the options, it's clear that option D, 10, 8, 6, 4, 2, ..., is the correct one. It starts with 10, which is our initial condition, and each subsequent term is 2 less than the previous one, just as our function dictates. This process of elimination is a powerful tool in mathematics. By understanding the properties of the function and the sequence it generates, we can quickly rule out incorrect options and focus on the one that aligns with our calculations. This also highlights the importance of carefully generating the first few terms of the sequence. These initial terms act as a fingerprint, allowing us to uniquely identify the correct sequence from a set of possibilities.

Conclusion

Great job, guys! We've successfully navigated the world of recursive functions and sequences. We started with a function, f(n+1) = f(n) - 2 with f(1) = 10, generated the sequence, and then identified the correct option. The sequence generated by the function is 10, 8, 6, 4, 2, ..., which corresponds to option D. This exercise demonstrates how we can use a recursive definition to create a sequence and how understanding the pattern of the sequence helps us to find the correct answer. Remember, in math, it's all about breaking down the problem, understanding the rules, and applying them step by step. Keep exploring, keep questioning, and keep learning!

I hope you found this explanation helpful and insightful. If you have any questions or want to explore other sequences, feel free to ask! Happy math-ing!