Finding The 12th Term: A Sequence Solution

by Alex Johnson 43 views

Have you ever stumbled upon a sequence and wondered how to pinpoint a specific term within it? Sequences are fundamental in mathematics, appearing in various contexts, from simple arithmetic progressions to more complex patterns. In this comprehensive guide, we'll delve into how to find the 12th term of a sequence, specifically when the sequence is defined by the formula an=−16+2na_n = -16 + 2n. We'll break down the formula, explain the steps involved, and provide clear examples to solidify your understanding. Whether you're a student tackling homework or simply curious about mathematical sequences, this article will equip you with the knowledge and skills to solve similar problems.

Understanding Sequences and Their Formulas

Let's start by understanding the basics. A sequence is an ordered list of numbers, often following a specific pattern or rule. Each number in the sequence is called a term, and these terms are usually indexed by their position in the sequence. For example, the first term is denoted as a1a_1, the second term as a2a_2, and so on. The general term, or the nth term, is represented as ana_n.

The formula an=−16+2na_n = -16 + 2n is a way to define a sequence explicitly. This means that we can directly calculate any term in the sequence by substituting the term number (n) into the formula. The formula tells us exactly how to find any term we desire without needing to know the preceding terms. This is particularly useful when seeking to find a term far down the sequence, such as the 12th term, as we'll explore in detail.

The variable 'n' represents the position of the term in the sequence. It's a natural number (1, 2, 3, ...) indicating whether we're looking for the first, second, third, or any subsequent term. The formula itself, −16+2n-16 + 2n, dictates the mathematical operation we perform on 'n' to arrive at the value of the term. In this case, we multiply 'n' by 2 and then subtract 16. This straightforward formula allows us to efficiently determine any term in this specific arithmetic sequence, making it a valuable tool for sequence analysis.

Understanding the structure and components of the formula is crucial before diving into calculations. It lays the foundation for grasping the behavior of the sequence and accurately determining any term within it. The ability to interpret such formulas is a cornerstone of mathematical proficiency, enabling one to solve a wide array of sequence-related problems. This foundational knowledge will prove invaluable as we proceed to find the 12th term of the given sequence, demonstrating the practical application of this theoretical understanding.

Step-by-Step Guide to Finding the 12th Term

Now, let's get down to business. Our goal is to find the 12th term (a12a_{12}) of the sequence defined by an=−16+2na_n = -16 + 2n. The process is quite straightforward, involving a simple substitution and calculation.

Step 1: Identify the Value of 'n'

Since we're looking for the 12th term, the value of 'n' is 12. This is the position of the term we want to find within the sequence. It's crucial to correctly identify 'n' as it forms the basis for our calculation. Misidentifying 'n' will lead to calculating the wrong term, highlighting the importance of this initial step. Recognizing that 'n' represents the term's position allows us to accurately apply the formula and determine the corresponding value in the sequence.

Step 2: Substitute 'n' into the Formula

Next, we substitute n = 12 into the formula an=−16+2na_n = -16 + 2n. This means we replace every instance of 'n' in the formula with the number 12. The equation now becomes a12=−16+2(12)a_{12} = -16 + 2(12). This substitution is the core of the process, transforming the general formula into a specific calculation for the 12th term. Ensuring accurate substitution is vital, as it directly impacts the final result. By carefully replacing 'n' with its numerical value, we set the stage for the arithmetic operations that will reveal the 12th term of the sequence.

Step 3: Perform the Calculation

Now, we perform the arithmetic calculation to find the value of a12a_{12}. Following the order of operations (PEMDAS/BODMAS), we first multiply 2 by 12, which equals 24. Our equation now looks like a12=−16+24a_{12} = -16 + 24. Then, we add -16 and 24, which results in 8. Therefore, a12=8a_{12} = 8. This final calculation is where the answer emerges, solidifying the value of the 12th term. Accuracy in arithmetic is paramount here; each step must be meticulously executed to ensure the correct outcome. The successful completion of this calculation provides the numerical answer, revealing the value of the 12th term in the sequence.

Conclusion: The 12th term of the sequence defined by an=−16+2na_n = -16 + 2n is 8. This step-by-step approach, involving identification of 'n', substitution into the formula, and careful calculation, can be applied to find any term in the sequence. Mastering this process allows for efficient and accurate determination of specific terms, highlighting the practical utility of sequence formulas in mathematical problem-solving. By breaking down the problem into manageable steps, we've demonstrated a clear methodology for tackling similar sequence-related questions, fostering a deeper understanding of mathematical sequences and their applications.

Example Problems and Solutions

To further solidify your understanding, let's work through a few example problems similar to the one we just solved. These examples will demonstrate how to apply the same steps to different sequences, reinforcing the methodology and showcasing its versatility.

Example 1:

Find the 10th term of the sequence defined by bn=5n−7b_n = 5n - 7.

  1. Identify the Value of 'n': We are looking for the 10th term, so n = 10.
  2. Substitute 'n' into the Formula: b10=5(10)−7b_{10} = 5(10) - 7
  3. Perform the Calculation: b10=50−7=43b_{10} = 50 - 7 = 43

Therefore, the 10th term of the sequence bn=5n−7b_n = 5n - 7 is 43. This example mirrors the previous problem's structure but utilizes a different formula. The key is to consistently follow the established steps: identify 'n', substitute it into the formula, and then perform the arithmetic. This methodical approach ensures accuracy and minimizes the risk of errors. By successfully solving this example, you're reinforcing your understanding of the core process and building confidence in your ability to tackle similar problems.

Example 2:

What is the 15th term in the sequence given by cn=−3+4nc_n = -3 + 4n?

  1. Identify the Value of 'n': We need to find the 15th term, so n = 15.
  2. Substitute 'n' into the Formula: c15=−3+4(15)c_{15} = -3 + 4(15)
  3. Perform the Calculation: c15=−3+60=57c_{15} = -3 + 60 = 57

Thus, the 15th term of the sequence cn=−3+4nc_n = -3 + 4n is 57. This example introduces a slight variation in the formula, with the constant term appearing before the term involving 'n'. However, the underlying process remains the same. By carefully substituting n = 15 and following the order of operations, we arrive at the correct answer. This reinforces the idea that the methodology is adaptable to different formula structures, highlighting its robustness and general applicability. Successfully navigating this example further solidifies your understanding and proficiency in handling sequence problems.

Example 3:

Determine the 20th term of the sequence defined by dn=−10+3nd_n = -10 + 3n.

  1. Identify the Value of 'n': We are looking for the 20th term, so n = 20.
  2. Substitute 'n' into the Formula: d20=−10+3(20)d_{20} = -10 + 3(20)
  3. Perform the Calculation: d20=−10+60=50d_{20} = -10 + 60 = 50

Hence, the 20th term of the sequence dn=−10+3nd_n = -10 + 3n is 50. This final example further reinforces the process, applying it to a sequence where 'n' is a larger number. This demonstrates that the method remains effective even when seeking terms farther down the sequence. By consistently applying the three steps, we can efficiently determine any term, regardless of its position. This successful completion of the example provides a conclusive demonstration of your mastery of the methodology, empowering you to confidently solve a wide range of sequence-related problems.

These examples illustrate the consistent application of the three-step process. By identifying 'n', substituting it into the formula, and performing the calculation, you can accurately find any term in a sequence defined by a similar formula. Practice with various sequences will further enhance your skills and build confidence in your ability to solve these types of problems. The key is to approach each problem systematically, ensuring accuracy in both substitution and calculation.

Common Mistakes and How to Avoid Them

While the process of finding a specific term in a sequence is relatively straightforward, certain common mistakes can lead to incorrect answers. Being aware of these pitfalls and knowing how to avoid them is crucial for ensuring accuracy in your calculations.

1. Incorrectly Identifying 'n':

One of the most frequent errors is misidentifying the value of 'n'. Remember, 'n' represents the position of the term you are trying to find in the sequence. For example, if you're asked to find the 12th term, n = 12. A common mistake is to confuse 'n' with the value of the term itself or another number provided in the problem statement. To avoid this, always reread the question carefully and explicitly identify the term number you are seeking. Underlining or highlighting the term number can be a helpful strategy to ensure you've correctly identified 'n'. This simple step can prevent a cascade of errors in subsequent calculations.

2. Errors in Substitution:

After identifying 'n', the next step is to substitute it into the formula. Errors in substitution often occur due to carelessness or a misunderstanding of the formula's structure. Ensure you replace every instance of 'n' in the formula with its numerical value. A common mistake is to only substitute 'n' in some parts of the formula, leaving others untouched. To prevent this, double-check your substitution, comparing the original formula with the substituted expression. A methodical approach, where you carefully replace each 'n' one at a time, can minimize the risk of errors. Accuracy in substitution is paramount, as even a small mistake can lead to a significantly different result.

3. Order of Operations Errors:

Once you've substituted 'n' into the formula, you need to perform the calculation. Errors in calculation often arise from not following the correct order of operations (PEMDAS/BODMAS). Remember to perform operations in the following order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). A common mistake is to perform addition or subtraction before multiplication, leading to an incorrect result. To avoid this, write out each step of the calculation clearly, ensuring you apply the order of operations correctly. Breaking down the calculation into smaller, manageable steps makes it easier to track your progress and identify any potential errors. Consistent adherence to PEMDAS/BODMAS is essential for accurate calculations.

4. Arithmetic Mistakes:

Even with the correct substitution and order of operations, simple arithmetic errors can occur. These can range from incorrect multiplication or division to mistakes in addition or subtraction. While these errors may seem minor, they can significantly impact the final answer. To minimize arithmetic mistakes, take your time and double-check your calculations. If the problem is complex, consider using a calculator to verify your results. A useful strategy is to perform the calculation twice, using different methods if possible, to ensure consistency. Paying close attention to detail and employing verification techniques can help catch and correct arithmetic errors.

By being aware of these common mistakes and implementing strategies to avoid them, you can significantly improve your accuracy in finding terms in sequences. Consistent practice and a methodical approach are key to mastering this skill. Remember, attention to detail and careful execution are essential for success in mathematical problem-solving.

Conclusion

In conclusion, finding the 12th term of a sequence, or any term for that matter, when given a formula like an=−16+2na_n = -16 + 2n, is a straightforward process once you understand the underlying concepts and follow a systematic approach. We've broken down the process into clear, manageable steps: identifying the value of 'n', substituting 'n' into the formula, and performing the calculation. By consistently applying these steps, you can confidently tackle similar problems involving sequences.

We also explored example problems to further solidify your understanding and showcase the versatility of the methodology. These examples demonstrated that the same three-step process can be applied to different sequence formulas, highlighting its general applicability. By working through these examples, you've gained practical experience in applying the concepts and building your problem-solving skills.

Furthermore, we addressed common mistakes that students often make when working with sequences and provided strategies for avoiding them. These included errors in identifying 'n', substitution mistakes, order of operations errors, and arithmetic inaccuracies. By being aware of these potential pitfalls and implementing the recommended preventative measures, you can significantly improve your accuracy and avoid common errors.

Mastering the skill of finding specific terms in sequences is a valuable asset in mathematics. It not only enhances your understanding of sequences but also develops your problem-solving abilities and attention to detail. The methodical approach we've outlined can be applied to a wide range of mathematical problems, making it a valuable tool in your mathematical toolkit. Continue to practice with different sequences and formulas to further refine your skills and build confidence in your ability to solve these types of problems.

For further exploration of sequences and series, you can visit Khan Academy's Sequences and Series section. This resource provides additional explanations, examples, and practice problems to help you deepen your understanding of this important mathematical topic.