Find The 66th Term: Arithmetic Sequence Explained

by Felix Dubois 50 views

Have you ever wondered how to find a specific term in a sequence of numbers? Perhaps you've stumbled upon an arithmetic sequence and felt a bit lost trying to figure out its pattern. Well, you're in the right place! In this guide, we'll dive deep into the world of arithmetic sequences, focusing on how to find the 66th term of the arithmetic sequence -28, -45, -62, ... We'll break down the concepts, walk through the steps, and provide you with the tools you need to tackle similar problems with confidence. So, let's get started, guys!

Understanding Arithmetic Sequences

Before we jump into solving our specific problem, let's first build a solid foundation by understanding what arithmetic sequences are all about. An arithmetic sequence, at its core, is a sequence of numbers where the difference between any two consecutive terms remains constant. This constant difference is what we call the "common difference." Think of it as a steady progression, where you're adding or subtracting the same value each time to get to the next term. This consistent pattern makes arithmetic sequences predictable and allows us to use formulas to find any term in the sequence without having to list out every single number.

To illustrate this, let's take a closer look at our sequence: -28, -45, -62, ... To confirm that this is indeed an arithmetic sequence, we need to check if the difference between consecutive terms is constant. Let's subtract the first term from the second term: -45 - (-28) = -17. Now, let's subtract the second term from the third term: -62 - (-45) = -17. As you can see, the difference is the same (-17) in both cases. This confirms that we are dealing with an arithmetic sequence, and our common difference (often denoted as d) is -17. This constant difference is the key to unlocking any term in the sequence, including the 66th term that we are after. Understanding this foundational concept is crucial, as it allows us to apply the appropriate formulas and techniques to solve our problem effectively. Without recognizing the constant difference, finding a specific term in an arithmetic sequence would be a much more challenging task.

Key Components of an Arithmetic Sequence

To effectively work with arithmetic sequences, it's essential to understand their key components. There are primarily three elements that define an arithmetic sequence:

  1. The First Term (a₁): This is the starting point of the sequence, the very first number in the list. In our sequence, -28, -45, -62, ..., the first term (a₁) is -28. It's the foundation upon which the entire sequence is built. Identifying the first term is crucial as it serves as a reference point for calculating subsequent terms. Without knowing where the sequence begins, it would be impossible to determine the position of any other term.

  2. The Common Difference (d): As we discussed earlier, the common difference is the constant value that is added (or subtracted) to each term to get the next term. In our example, we found that the common difference (d) is -17. This value dictates the rate at which the sequence progresses. A positive common difference indicates an increasing sequence, while a negative common difference indicates a decreasing sequence. The common difference is the heartbeat of the arithmetic sequence, driving its pattern and allowing us to predict future terms.

  3. The nth Term (an): This represents the term at a specific position n in the sequence. For example, the 3rd term would be denoted as a₃. Our goal is to find the 66th term, which would be represented as a₆₆. The nth term is the target we aim to reach, the specific number in the sequence that we want to identify. It could be any term, from the 1st to the 100th, or even the 1000th, depending on the problem we're trying to solve. Understanding the concept of the nth term is vital for solving a wide range of arithmetic sequence problems.

Understanding these components is like having the ingredients for a recipe. Once you know the first term, the common difference, and what term you're looking for, you have all the information you need to find your answer. These elements work together to define the sequence and allow us to use a powerful formula to calculate any term we desire.

The Formula for the nth Term

Now that we have a solid understanding of arithmetic sequences and their components, it's time to introduce the key tool that will help us find any term in the sequence: the formula for the nth term. This formula is a powerful shortcut that allows us to bypass the tedious process of listing out every term until we reach the one we're looking for. Instead, we can simply plug in the relevant values and calculate the result directly. The formula for the nth term (aₙ) of an arithmetic sequence is given by:

aₙ = a₁ + (n - 1)d

Let's break down what each part of this formula means:

  • aₙ: This is the nth term that we want to find. It's the unknown value that we're trying to calculate.
  • a₁: This is the first term of the sequence, as we discussed earlier. It's our starting point.
  • n: This is the position of the term we want to find. For example, if we want the 10th term, n would be 10.
  • d: This is the common difference between consecutive terms, the constant value that's added or subtracted.

This formula is based on the fundamental principle of arithmetic sequences: each term is obtained by adding the common difference to the previous term. The (n - 1) part of the formula accounts for the number of times we need to add the common difference to the first term to reach the nth term. For instance, to get to the 5th term, we need to add the common difference 4 times (5 - 1 = 4) to the first term. This is the magic behind the formula, allowing us to jump directly to any term without having to step through each one individually.

Applying the Formula to Our Problem

Now comes the exciting part: applying the formula to find the 66th term of the arithmetic sequence -28, -45, -62, ... We've already identified the key components of our sequence:

  • a₁ = -28 (the first term)
  • d = -17 (the common difference)
  • n = 66 (the term we want to find)

We have all the ingredients we need! Now, let's plug these values into our formula:

a₆₆ = a₁ + (n - 1)d a₆₆ = -28 + (66 - 1)(-17)

Now, let's simplify the equation step by step:

a₆₆ = -28 + (65)(-17) a₆₆ = -28 + (-1105) a₆₆ = -1133

And there you have it! The 66th term of the arithmetic sequence -28, -45, -62, ... is -1133. We've successfully used the formula to bypass the need to list out all 66 terms and arrived at our answer directly. This demonstrates the power and efficiency of the formula for the nth term. It's a valuable tool for anyone working with arithmetic sequences, allowing us to solve problems quickly and accurately.

Step-by-Step Solution

To further solidify your understanding, let's walk through the solution step-by-step:

  1. Identify the First Term (a₁): In our sequence, -28, -45, -62, ..., the first term is clearly -28. This is our starting point, the foundation upon which the sequence is built. It's the first number in the list, and we'll use it as a reference point for calculating subsequent terms.

  2. Determine the Common Difference (d): To find the common difference, we subtract any term from the term that follows it. Let's subtract the first term from the second term: -45 - (-28) = -17. We can confirm this by subtracting the second term from the third term: -62 - (-45) = -17. Since the difference is consistent, our common difference is -17. This constant difference is the key to the arithmetic sequence, dictating how the sequence progresses.

  3. Identify the Term You Want to Find (n): We are asked to find the 66th term, so n = 66. This is the position of the term we're interested in. It tells us how far along the sequence we need to go to find our target number.

  4. Apply the Formula for the nth Term: Now that we have all the necessary components, we can plug them into the formula: aₙ = a₁ + (n - 1)d. Substituting our values, we get: a₆₆ = -28 + (66 - 1)(-17).

  5. Simplify the Equation: Let's simplify the equation step by step:

    • a₆₆ = -28 + (65)(-17)
    • a₆₆ = -28 + (-1105)
    • a₆₆ = -1133

  6. State the Answer: Therefore, the 66th term of the arithmetic sequence -28, -45, -62, ... is -1133. We've successfully navigated through the steps, applying the formula and arriving at our final answer. This step-by-step approach provides a clear and organized method for solving arithmetic sequence problems.

Practice Problems

To truly master the art of finding terms in arithmetic sequences, practice is key! Here are a few problems for you to try:

  1. Find the 40th term of the arithmetic sequence 5, 11, 17, ...
  2. What is the 100th term of the arithmetic sequence 2, -1, -4, ...?
  3. Determine the 25th term of the arithmetic sequence -10, -4, 2, ...

By working through these problems, you'll not only reinforce your understanding of the formula but also develop your problem-solving skills. Remember to identify the first term, the common difference, and the term you want to find, then plug those values into the formula. With consistent practice, you'll become a pro at finding any term in an arithmetic sequence!

Conclusion

In this guide, we've explored the fascinating world of arithmetic sequences and learned how to find the 66th term of the arithmetic sequence -28, -45, -62, ... We started by understanding the basic components of an arithmetic sequence: the first term, the common difference, and the nth term. Then, we introduced the powerful formula for the nth term: aₙ = a₁ + (n - 1)d. We walked through the step-by-step solution, demonstrating how to apply the formula to our specific problem. Finally, we provided practice problems to help you solidify your understanding and develop your skills.

Arithmetic sequences are a fundamental concept in mathematics, and mastering them opens doors to more advanced topics. The ability to find a specific term in an arithmetic sequence is a valuable skill that can be applied in various real-world scenarios, from predicting patterns to solving financial problems. So, keep practicing, keep exploring, and keep unlocking the secrets of mathematics! You've got this, guys!