Explicit Formula For Sequence 3, 7, 11, 15
Hey guys! Today, we're diving into a common type of math problem: finding the explicit formula for a sequence. Specifically, we'll be looking at the sequence 3, 7, 11, and 15. We're aiming to determine a formula, f(n), that will directly give us any term in the sequence based on its position, n. This is super useful because instead of listing out terms to find, say, the 100th term, we can just plug 100 into our formula. Let's get started and break down how to approach this kind of problem, making sure we really understand the steps involved. It's not just about finding the answer, but about understanding the why behind it. So, grab your thinking caps, and let's get to it!
Understanding Arithmetic Sequences
Before we jump into finding the formula, let's chat a bit about what kind of sequence we're dealing with here. Looking at the numbers 3, 7, 11, and 15, you might notice a pattern. Each term is a fixed amount larger than the one before it. This, my friends, is the hallmark of an arithmetic sequence. An arithmetic sequence is simply a sequence where the difference between any two consecutive terms is constant. This constant difference is often called the common difference. In our case, if we subtract any term from the one following it (like 7 - 3, or 11 - 7), we get 4. So, our common difference is 4. Identifying this common difference is a crucial first step because it directly relates to the structure of our explicit formula. Remember, recognizing patterns is key in math, and arithmetic sequences are one of the most fundamental patterns you'll encounter. Understanding this concept deeply will not only help with this specific problem but also with many other areas of mathematics. We're building a foundation here, folks!
Determining the Common Difference
The common difference is the backbone of an arithmetic sequence, and figuring it out is usually our first order of business. As we touched on earlier, the common difference is the constant value that's added to each term to get the next one. For our sequence (3, 7, 11, 15), we can easily spot this by subtracting consecutive terms. Let's do it explicitly: 7 - 3 = 4, 11 - 7 = 4, and 15 - 11 = 4. See? It's consistently 4. Therefore, the common difference, which we often denote as d, is 4. This d value is super important because it's going to play a direct role in our explicit formula. Think of it as the 'growth rate' of the sequence – how much the sequence increases (or decreases, if d were negative) with each step. Mastering the identification of the common difference is a foundational skill for working with arithmetic sequences, so make sure you're comfortable with this step. It’s like learning the alphabet before you can write words; you gotta have the basics down!
Constructing the Explicit Formula
Okay, now for the fun part – building the explicit formula! The explicit formula allows us to calculate any term in the sequence directly, without having to list out all the preceding terms. For arithmetic sequences, the general form of the explicit formula is:
f(n) = a + (n - 1)d
Where:
- f(n) is the nth term of the sequence (the term we want to find)
- a is the first term of the sequence
- n is the position of the term in the sequence (1 for the first term, 2 for the second, and so on)
- d is the common difference
We've already identified d as 4 for our sequence. We can also see that the first term, a, is 3. So, let's plug these values into our general formula:
f(n) = 3 + (n - 1)4
Now, we can simplify this a bit by distributing the 4:
f(n) = 3 + 4n - 4
And combining like terms:
f(n) = 4n - 1
Boom! We've got our explicit formula. This formula tells us exactly how to find any term in the sequence 3, 7, 11, 15. It's like having a magic key that unlocks any term we want. Let's test it out to make sure it works, just to be extra sure!
Verifying the Formula
Alright, guys, before we pat ourselves on the back, let's verify that our formula, f(n) = 4n - 1, actually works. This is a crucial step in problem-solving – always double-check your work! To do this, we'll plug in some values for n (the term number) and see if the formula gives us the corresponding term in the sequence. Let's try the first few terms:
- For the first term (n = 1): f(1) = 4(1) - 1 = 4 - 1 = 3. This matches the first term in our sequence.
- For the second term (n = 2): f(2) = 4(2) - 1 = 8 - 1 = 7. This matches the second term.
- For the third term (n = 3): f(3) = 4(3) - 1 = 12 - 1 = 11. This matches the third term.
- For the fourth term (n = 4): f(4) = 4(4) - 1 = 16 - 1 = 15. And this matches the fourth term!
Excellent! Our formula holds up for the terms we know. This gives us a high degree of confidence that it's correct. Verification is not just a formality; it's about building confidence in your solution and catching any potential errors. It’s like proofreading an essay before you submit it – you want to make sure everything is solid.
Choosing the Correct Option
Now that we've derived and verified our explicit formula, f(n) = 4n - 1, we can confidently choose the correct option from the given choices. Looking back at the options:
A. f(n) = 4n - 1 B. f(n) = n + 4 C. f(n) = 4n + 3 D. f(n) = 3n - 4
It's clear that option A matches our derived formula. So, f(n) = 4n - 1 is the explicit formula that represents the sequence 3, 7, 11, 15. This final step is about making sure you're answering the question directly and clearly. It’s like putting the final piece of the puzzle in place – it completes the picture.
Conclusion
So, guys, we did it! We successfully found the explicit formula for the sequence 3, 7, 11, 15. We started by recognizing that it's an arithmetic sequence, identified the common difference, built the explicit formula, verified it, and then confidently chose the correct option. Remember, the key takeaways here are:
- Identifying arithmetic sequences by their common difference.
- Understanding the general form of the explicit formula: f(n) = a + (n - 1)d.
- The importance of verifying your solution.
These skills are not just for this problem; they're fundamental to understanding and working with sequences and series in mathematics. Keep practicing, keep exploring, and you'll become a master of these concepts in no time! Remember, math is like building a tower – each concept you learn is a block that makes the tower stronger and taller. Keep stacking those blocks!