Unveiling Arithmetic Sequences: Formulas And Solutions

Hey guys! Let's dive into the fascinating world of arithmetic sequences. We'll explore how to crack those formulas and figure out the missing pieces. We'll look at both explicit and recursive formulas, so you'll be a sequence pro in no time! This particular problem is about understanding how arithmetic sequences work, both in their explicit and recursive forms. Arithmetic sequences are super important in math and pop up in all sorts of places.

Understanding Arithmetic Sequences: The Basics

So, what exactly is an arithmetic sequence? Well, it's a list of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference. Think of it like climbing stairs – you always go up (or down) by the same amount each step. Let's say we have a sequence: 2, 5, 8, 11, 14... See it? The common difference is 3 because you add 3 to get from one number to the next. Understanding this constant difference is absolutely key to working with these sequences. The common difference is the secret ingredient! This concept is the foundation for all the formulas and problem-solving we are about to do. You will also understand how useful it is for quickly calculating terms in the sequence without having to list them all out. Furthermore, we'll look at how to switch between the two main ways of writing down an arithmetic sequence: explicit and recursive formulas. They both give you the same sequence, but they look a little different, and they're useful in different ways, so you'll be a pro at using them both. If you're dealing with a sequence that's not arithmetic, meaning the difference isn't constant, you can't use these formulas directly. You'll need a different set of tools, which we can get into if you'd like. But, for now, we're sticking to the nice, predictable world of arithmetic sequences. Arithmetic sequences are used to model real-world situations. They are also used in financial calculations, such as calculating the interest earned on an investment. In fact, some basic computer algorithms are also based on arithmetic sequences.

Explicit Formula: Jumping to Any Term

The explicit formula is like a shortcut to find any term in the sequence without having to list out all the terms before it. It's a direct formula, which is why it's called explicit. The general form of the explicit formula is: a_n = a_1 + (n - 1) * d, where: a_n is the nth term, a_1 is the first term, n is the term number (e.g., 1 for the first term, 2 for the second term, etc.), and d is the common difference. Let's look back at our example sequence: 2, 5, 8, 11, 14... If we want to find the 10th term (a_10), we'd plug in the values: a_10 = 2 + (10 - 1) * 3 = 2 + 9 * 3 = 2 + 27 = 29. So, the 10th term is 29. Pretty neat, huh? It's all about knowing the first term and the common difference. The explicit formula is super useful when you need to find a term far out in the sequence. It saves a bunch of time compared to writing out every single term until you get to the one you want. It is a handy tool, especially when dealing with larger values of n. Understanding how the explicit formula works helps in building a solid foundation for solving sequence problems. And, as a bonus, the explicit formula gives you a clear understanding of how each term relates to its position in the sequence. The explicit formula provides a straightforward way to pinpoint any term in an arithmetic sequence. This contrasts with the recursive formula, which needs you to know the previous term. With the explicit formula, you just plug in the 'n' value, and boom, you have your answer!

Recursive Formula: Step-by-Step

The recursive formula is like giving directions to get to the next step. It tells you how to get the current term based on the previous term. The general form of a recursive formula is: a_n = a_{n-1} + d, where: a_n is the nth term, a_{n-1} is the previous term, and d is the common difference. In our example sequence (2, 5, 8, 11, 14...), the recursive formula would be: a_n = a_{n-1} + 3. This means to get any term, you add 3 to the previous term. If you know that the first term, a_1, is 2, then the second term, a_2, is 2 + 3 = 5, the third term, a_3, is 5 + 3 = 8, and so on. The recursive formula is great when you need to calculate a few consecutive terms, but it can be less efficient if you're looking for a term way down the line. You would need to calculate all the terms before it. Recursive formulas are great for modeling processes where each step depends on the one before it. They highlight the relationship between consecutive terms, emphasizing the iterative nature of the sequence. The formula shows that each term is built from the previous one, showing the step-by-step pattern of the sequence. You can think of the recursive formula as building a staircase. Each step (term) is constructed based on the one before it, adding the same height (common difference) to get to the next level. The beauty of the recursive formula lies in its simplicity and its ability to show the fundamental building block of the sequence. However, using the recursive formula to calculate a term far down the line can be a bit of a drag. You gotta know all the terms before it. This makes the explicit formula more appealing for finding terms deep into the sequence.

Solving the Problem: Finding the Missing Piece

Alright, let's get to the problem! We're given an explicit formula and asked to find the missing piece in the recursive formula. Here's what we know:

• Explicit Formula: a_n = 13 + (n - 1) * 6
• Recursive Formula: a_n = a_{n-1} + ?

First, we need to figure out the common difference from the explicit formula. Remember, the explicit formula is a_n = a_1 + (n - 1) * d. Comparing the given explicit formula (a_n = 13 + (n - 1) * 6) to the general form, we can see that the common difference (d) is 6. That's the number being multiplied by (n - 1). The common difference is the key! It's what we add to the previous term to get the next term. So, the missing piece in the recursive formula is the common difference, which is 6. The recursive formula becomes: a_n = a_{n-1} + 6. The answer is D. 6!

Breaking Down the Solution

Let's break down how we got there. The explicit formula tells us the direct relationship between the term number (n) and the value of the term (a_n). The + (n-1) * 6 part means that for every step forward in the sequence, we add 6. That '6' is the common difference. It represents the constant amount we add to each term to get the next one. When we use the recursive formula, we're saying,