Finding The 7th Term Of A Geometric Sequence 6, 12, 24 A Step By Step Guide

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  • Introduction
  • Understanding Geometric Sequences
  • Identifying the Pattern
  • Calculating the 7th Term
  • Detailed Solution
  • Why 384 is the Correct Answer
  • Common Mistakes to Avoid
  • Practice Questions
  • Real-World Applications of Geometric Sequences
  • Conclusion

Introduction

In the realm of mathematics, sequences and series form a fundamental concept, with geometric sequences holding a special place due to their consistent multiplicative pattern. These sequences not only serve as the backbone for various mathematical theories but also find practical applications in diverse fields such as finance, physics, and computer science. One common problem involves finding a specific term in a geometric sequence, which requires a clear understanding of the sequence's underlying pattern. In this article, we will delve into a particular geometric sequence: 6, 12, 24, ..., and systematically determine its 7th term. This exploration will involve a step-by-step approach, highlighting the core principles of geometric sequences and the methods used to solve related problems. By the end of this discussion, you will not only know the answer but also grasp the methodology to tackle similar challenges. Let's embark on this mathematical journey to uncover the 7th term of this intriguing sequence.

Understanding Geometric Sequences

To effectively tackle the problem at hand, it's crucial to first understand the basics of geometric sequences. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This constant ratio is what distinguishes geometric sequences from arithmetic sequences, where terms are found by adding a constant difference. The general form of a geometric sequence can be expressed as: a, ar, ar², ar³, ..., where 'a' is the first term and 'r' is the common ratio. Understanding this foundational concept is essential because it provides the framework for identifying patterns and predicting future terms in the sequence. When you encounter a sequence, the first step is often to determine if it is geometric by checking if there is a consistent ratio between consecutive terms. This involves dividing any term by its preceding term and ensuring that the result is the same across the sequence. This common ratio is the key to unlocking the behavior of the sequence and finding any term within it. Once you've identified the common ratio, you can use the formula for the nth term of a geometric sequence to calculate any term in the series. This formula is a powerful tool that makes it possible to predict values far into the sequence without having to manually compute each term. Thus, a solid grasp of geometric sequences is not just about memorizing the formula but understanding the underlying multiplicative pattern that governs their structure.

Identifying the Pattern

The first step in solving this problem is to identify the pattern within the given sequence: 6, 12, 24, .... To determine if this is a geometric sequence, we need to check if there is a common ratio between consecutive terms. We can do this by dividing each term by its preceding term. Let's start by dividing the second term (12) by the first term (6): 12 / 6 = 2. Next, we divide the third term (24) by the second term (12): 24 / 12 = 2. Since the ratio between consecutive terms is consistently 2, we can confirm that this is indeed a geometric sequence with a common ratio (r) of 2. The first term (a) of the sequence is 6. Now that we have identified the first term and the common ratio, we have the necessary components to find any term in the sequence. Understanding the common ratio is crucial because it dictates how the sequence progresses. In this case, each term is twice the previous term, which indicates exponential growth. Recognizing this pattern allows us to use the general formula for the nth term of a geometric sequence, which will be essential for calculating the 7th term. By carefully analyzing the sequence and confirming the common ratio, we have laid the foundation for solving the problem efficiently and accurately. This initial step of pattern identification is not only critical for this specific question but also for tackling any problem involving sequences and series. It's a fundamental skill that helps in recognizing mathematical structures and predicting future values.

Calculating the 7th Term

Now that we've established that the sequence is geometric with a first term (a) of 6 and a common ratio (r) of 2, we can proceed to calculate the 7th term. The formula for the nth term of a geometric sequence is given by: a_n = a * r^(n-1), where a_n is the nth term, a is the first term, r is the common ratio, and n is the term number we want to find. In our case, we want to find the 7th term (n = 7), so we will substitute the known values into the formula: a_7 = 6 * 2^(7-1). First, we simplify the exponent: 7 - 1 = 6. So the equation becomes: a_7 = 6 * 2^6. Next, we calculate 2^6, which is 2 * 2 * 2 * 2 * 2 * 2 = 64. Now, we multiply 6 by 64: a_7 = 6 * 64. Performing this multiplication gives us: a_7 = 384. Therefore, the 7th term of the geometric sequence 6, 12, 24, ... is 384. This systematic calculation demonstrates the power of the formula for the nth term, allowing us to efficiently find any term in the sequence without having to manually compute each preceding term. The formula encapsulates the essence of geometric growth, where each term is a multiple of the previous term, determined by the common ratio. By accurately applying this formula and performing the necessary arithmetic, we have successfully determined the 7th term, which aligns with option a) 384.

Detailed Solution

To ensure clarity and provide a comprehensive understanding, let's outline a detailed solution for finding the 7th term of the series 6, 12, 24, ....

  1. Identify the Sequence Type:

    • First, we examine the sequence 6, 12, 24, ... to determine if it's geometric. A geometric sequence is characterized by a constant ratio between consecutive terms.

  2. Calculate the Common Ratio (r):

    • To find the common ratio, we divide the second term by the first term: 12 / 6 = 2.
    • We then divide the third term by the second term: 24 / 12 = 2. Since the ratio is consistent, the common ratio (r) is 2.

  3. Identify the First Term (a):

    • The first term of the sequence is 6. So, a = 6.

  4. Apply the Formula for the nth Term:

    • The formula for the nth term of a geometric sequence is a_n = a * r^(n-1).

  5. Substitute the Values:

    • We want to find the 7th term (n = 7), so we substitute a = 6, r = 2, and n = 7 into the formula:
      • a_7 = 6 * 2^(7-1)

  6. Simplify the Exponent:

    • Subtract 1 from 7: 7 - 1 = 6.
      • The equation becomes: a_7 = 6 * 2^6

  7. Calculate 2^6:

    • 2^6 = 2 * 2 * 2 * 2 * 2 * 2 = 64

  8. Multiply by the First Term:

    • Multiply 6 by 64: a_7 = 6 * 64

  9. Find the 7th Term:

    • a_7 = 384

Therefore, the 7th term of the series 6, 12, 24, ... is 384. This detailed breakdown illustrates each step involved in solving the problem, from identifying the sequence type to applying the formula and calculating the result. By understanding each step, you can confidently tackle similar problems involving geometric sequences. This methodical approach ensures accuracy and clarity, making the solution process easier to follow and comprehend.

Why 384 is the Correct Answer

To solidify our understanding, let's explore in detail why 384 is the correct answer for the 7th term of the geometric sequence 6, 12, 24, .... We have already established that the sequence is geometric with a first term (a) of 6 and a common ratio (r) of 2. This means each term is obtained by multiplying the previous term by 2. To find the 7th term, we use the formula a_n = a * r^(n-1). Substituting the values, we get a_7 = 6 * 2^(7-1) = 6 * 2^6. We calculated 2^6 as 64, and then multiplied it by 6 to get 384. This result aligns perfectly with the properties of the geometric sequence. Let's list out the first few terms to further illustrate this: 1st term: 6 2nd term: 6 * 2 = 12 3rd term: 12 * 2 = 24 4th term: 24 * 2 = 48 5th term: 48 * 2 = 96 6th term: 96 * 2 = 192 7th term: 192 * 2 = 384 As we can see, the 7th term indeed equals 384, confirming our calculation. The geometric progression ensures a consistent pattern of growth, which is clearly evident in this sequence. Each term doubles, and this consistent multiplication leads us directly to 384 as the 7th term. This step-by-step verification reinforces the accuracy of our solution and demonstrates the reliability of the formula for the nth term of a geometric sequence. By understanding the underlying pattern and applying the formula correctly, we can confidently determine any term in the sequence, making 384 the definitive answer for the 7th term.

Common Mistakes to Avoid

When working with geometric sequences, there are several common mistakes to avoid to ensure accurate solutions. Understanding these pitfalls can significantly improve your problem-solving skills. One frequent error is incorrectly identifying the common ratio. It's crucial to consistently divide each term by its preceding term to verify the ratio. If the ratio isn't constant, then the sequence isn't geometric. Another mistake is confusing geometric sequences with arithmetic sequences, where terms increase by a constant difference rather than a constant ratio. Always check the pattern carefully before applying formulas. A crucial error occurs when using the formula for the nth term (a_n = a * r^(n-1)). A common mistake is to calculate r^n instead of r^(n-1). Remember that the exponent should be one less than the term number you are trying to find. Arithmetic errors in calculations are also common. Make sure to perform each step carefully, especially when dealing with exponents and multiplication. It's always a good practice to double-check your calculations to avoid simple mistakes. Another mistake to watch out for is misunderstanding the first term (a). Ensure you correctly identify the first term in the sequence, as it is the foundation for calculating subsequent terms. Finally, it's essential to understand the concept of geometric sequences thoroughly. Rote memorization of formulas without understanding the underlying principles can lead to errors. Make sure you grasp the concept of a common ratio and how it drives the sequence's progression. By being mindful of these common mistakes, you can enhance your accuracy and confidence when solving problems involving geometric sequences.

Practice Questions

To reinforce your understanding of geometric sequences and the methods for finding specific terms, let's explore some practice questions. These exercises will help you apply the concepts we've discussed and refine your problem-solving skills.

  1. Question 1:

    • Find the 6th term of the geometric sequence: 2, 6, 18, ...

  2. Question 2:

    • What is the 9th term of the geometric sequence: 1, -3, 9, ...?

  3. Question 3:

    • Determine the 5th term of the geometric sequence: 5, 10, 20, ...

  4. Question 4:

    • Calculate the 8th term of the geometric sequence: 3, 6, 12, ...

  5. Question 5:

    • Find the 7th term of the geometric sequence: 4, -8, 16, ...

These practice questions cover a range of geometric sequences with varying first terms and common ratios, allowing you to apply the formula for the nth term in different scenarios. By working through these problems, you will become more adept at identifying geometric sequences, calculating common ratios, and using the formula effectively. It's recommended to solve each question step-by-step, showing your work, to ensure you understand the process thoroughly. After attempting the questions, you can compare your answers with solutions to verify your understanding and identify any areas where you may need further practice. These exercises are crucial for solidifying your knowledge and building confidence in your ability to tackle geometric sequence problems.

Real-World Applications of Geometric Sequences

Geometric sequences are not just theoretical mathematical concepts; they have numerous real-world applications that make them incredibly useful in various fields. Understanding these applications can provide a deeper appreciation for the practical significance of geometric sequences. One common application is in finance, particularly in calculating compound interest. The growth of an investment with compound interest follows a geometric sequence, where the principal amount is multiplied by a fixed interest rate each compounding period. This allows investors to predict the future value of their investments over time. In population growth, geometric sequences can model the exponential increase in population size under ideal conditions. If a population doubles every certain period, the population sizes at those intervals form a geometric sequence. Radioactive decay is another area where geometric sequences play a crucial role. The amount of a radioactive substance decreases by a fixed fraction over equal time intervals, which can be modeled using a geometric sequence. In computer science, geometric sequences are used in algorithms and data structures. For example, binary search algorithms divide the search space in half at each step, forming a geometric sequence of remaining elements. Geometric sequences also appear in physics, such as in the motion of a bouncing ball. The height of each subsequent bounce is a fraction of the previous one, creating a geometric sequence of bounce heights. These examples highlight the versatility of geometric sequences in modeling and understanding various real-world phenomena. By recognizing these patterns, we can apply mathematical principles to solve practical problems in diverse fields.

Conclusion

In conclusion, understanding geometric sequences is essential not only for mathematical proficiency but also for recognizing and solving problems in various real-world scenarios. Throughout this article, we have systematically explored the concept of geometric sequences, focusing on the specific problem of finding the 7th term of the series 6, 12, 24, .... We began by defining geometric sequences and highlighting their unique multiplicative pattern. We then identified the pattern in the given sequence, confirming that it is geometric with a first term of 6 and a common ratio of 2. By applying the formula for the nth term of a geometric sequence (a_n = a * r^(n-1)), we calculated the 7th term to be 384, providing a detailed step-by-step solution for clarity. We further reinforced this understanding by explaining why 384 is the correct answer and discussing common mistakes to avoid when working with geometric sequences. The practice questions provided an opportunity to apply the learned concepts, enhancing problem-solving skills. Finally, we explored the real-world applications of geometric sequences, showcasing their relevance in fields such as finance, population growth, radioactive decay, computer science, and physics. By mastering the concepts and techniques discussed in this article, you are well-equipped to tackle a wide range of problems involving geometric sequences. The ability to identify patterns, apply formulas, and understand real-world implications is invaluable in both academic and practical settings. As you continue your mathematical journey, remember that practice and a solid understanding of fundamental concepts are the keys to success. Geometric sequences, with their elegant simplicity and wide applicability, are a testament to the power and beauty of mathematics.