Calculating The 12th Term Of A Geometric Sequence A Comprehensive Guide

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In the realm of mathematics, sequences play a pivotal role in understanding patterns and progressions. Among these, geometric sequences hold a special significance due to their consistent multiplicative nature. A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a constant factor, known as the common ratio. This fundamental concept has applications across various fields, from finance and computer science to physics and biology. In this article, we will delve into the intricacies of geometric sequences and provide a step-by-step guide on how to calculate the 12th term of the sequence 1, 2, 4, 8, 16,...

To fully grasp the concept, let's consider a real-world example. Imagine you deposit $1,000 into a savings account that earns 5% interest compounded annually. The balance at the end of each year forms a geometric sequence. The initial deposit is the first term, and the interest rate determines the common ratio. Understanding how to calculate terms in a geometric sequence allows you to project the growth of your investment over time. Similarly, in computer science, geometric sequences are used in analyzing the efficiency of algorithms, where the number of operations may increase geometrically with the size of the input. In physics, radioactive decay follows a geometric pattern, with the amount of radioactive material decreasing by a constant factor over time. These examples highlight the diverse applications of geometric sequences and underscore the importance of mastering their underlying principles. Let's embark on a journey to compute the 12th term of a given geometric sequence, unraveling the elegance and practicality of this mathematical concept.

Before we can calculate the 12th term of the given sequence, it's essential to identify the key components that define a geometric sequence. These components are the first term (a1{a_1}), the common ratio (r), and the term number (n). The first term is simply the initial value of the sequence. In our case, the sequence starts with 1, so a1=1{a_1 = 1}. The common ratio is the constant factor by which each term is multiplied to obtain the next term. To find the common ratio, we can divide any term by its preceding term. For instance, dividing the second term (2) by the first term (1) gives us 2. Similarly, dividing the third term (4) by the second term (2) also yields 2. This confirms that the common ratio (r) for this sequence is 2.

The term number (n) represents the position of a term in the sequence. We are tasked with finding the 12th term, so n = 12. With these key components identified, we are well-equipped to apply the formula for finding the nth term of a geometric sequence. Understanding these components is crucial because they provide the foundation for applying the geometric sequence formula. The first term sets the starting point, the common ratio dictates the rate of growth or decay, and the term number specifies which term we are interested in. Without correctly identifying these components, our calculation of the 12th term would be inaccurate. For example, if we mistakenly identified the common ratio as 3 instead of 2, our subsequent calculations would lead to a completely different result. Therefore, a thorough understanding of these components is paramount to solving problems involving geometric sequences.

The cornerstone of calculating any term in a geometric sequence is the formula for the nth term. This formula provides a direct way to find any term in the sequence without having to calculate all the preceding terms. The formula is expressed as: an=a1∗r(n−1){a_n = a_1 * r^(n-1)} Where:

  • (an{a_n}) is the nth term (the term we want to find).
  • (a1{a_1}) is the first term of the sequence.
  • (r) is the common ratio.
  • (n) is the term number.

This formula elegantly captures the multiplicative nature of geometric sequences. It states that the nth term is equal to the first term multiplied by the common ratio raised to the power of (n-1). The exponent (n-1) reflects the fact that the first term is not multiplied by the common ratio, but each subsequent term is. To illustrate the power of this formula, let's consider a scenario where we need to find the 100th term of a geometric sequence. Calculating the first 99 terms and then multiplying by the common ratio would be a tedious and time-consuming process. However, by using the formula, we can directly compute the 100th term with just a few simple calculations. This formula is not just a mathematical tool; it is a gateway to understanding the behavior of geometric sequences and their applications in various fields. For instance, in finance, this formula can be used to calculate the future value of an investment that grows at a constant rate. In computer science, it can help determine the time complexity of certain algorithms. Mastering this formula is essential for anyone seeking to explore the fascinating world of geometric sequences and their real-world implications.

Now that we have identified the key components and introduced the formula for the nth term, let's apply this knowledge to calculate the 12th term of the geometric sequence 1, 2, 4, 8, 16, ... Here's a step-by-step breakdown of the calculation:

  1. Identify the values:
    • (a1{a_1} = 1) (the first term)
    • (r = 2) (the common ratio)
    • (n = 12) (the term number)
  2. Substitute the values into the formula: a12=a1∗r(n−1){a_{12} = a_1 * r^(n-1)} a12=1∗2(12−1){a_{12} = 1 * 2^(12-1)}
  3. Simplify the exponent: a12=1∗2(11){a_{12} = 1 * 2^(11)}
  4. Calculate 2 raised to the power of 11: 2(11)=2048{2^(11) = 2048}
  5. Multiply by the first term: a12=1∗2048{a_{12} = 1 * 2048} a12=2048{a_{12} = 2048}

Therefore, the 12th term of the geometric sequence is 2048. This step-by-step calculation demonstrates the power and efficiency of the formula. By simply substituting the known values into the formula and performing the necessary arithmetic operations, we were able to find the 12th term without having to calculate the preceding terms. This methodical approach is not only accurate but also scalable. Whether we are calculating the 12th term or the 100th term, the process remains the same. The key is to correctly identify the first term, the common ratio, and the term number, and then apply the formula diligently. This step-by-step approach also highlights the importance of understanding the order of operations in mathematics. The exponent is calculated before multiplication, ensuring that we arrive at the correct result. By breaking down the calculation into manageable steps, we minimize the risk of errors and gain a deeper understanding of the underlying mathematical principles.

To further solidify your understanding, let's delve into a more detailed explanation of each step involved in calculating the 12th term. This breakdown will not only reinforce the application of the formula but also address any potential areas of confusion.

  • Step 1: Identify the values: This initial step is crucial for setting the stage for the calculation. We begin by carefully examining the given sequence and extracting the necessary information. The first term, (a1{a_1}), is the starting point of the sequence, which in this case is 1. The common ratio, (r), is the constant factor that multiplies each term to produce the next. We determine this by dividing any term by its preceding term (e.g., 2/1 = 2, 4/2 = 2). The term number, (n), represents the position of the term we want to find, which is 12 in this problem. Correctly identifying these values is paramount, as any error at this stage will propagate through the rest of the calculation.
  • Step 2: Substitute the values into the formula: Once we have identified the values, the next step is to plug them into the formula for the nth term: an=a1∗r(n−1){a_n = a_1 * r^(n-1)} Replacing the variables with their corresponding values, we get: a12=1∗2(12−1){a_{12} = 1 * 2^(12-1)} This substitution transforms the abstract formula into a concrete equation specific to our problem. It's like fitting the pieces of a puzzle together, where each value plays a crucial role in determining the final result.
  • Step 3: Simplify the exponent: Before we can calculate the power, we need to simplify the exponent. In this case, we have (12-1), which simplifies to 11. This step is a simple application of arithmetic, but it's essential for maintaining the correct order of operations. Our equation now looks like this: a12=1∗2(11){a_{12} = 1 * 2^(11)} Simplifying the exponent makes the next step, calculating the power, more manageable.
  • Step 4: Calculate 2 raised to the power of 11: This is where we calculate the value of (2^11). This means multiplying 2 by itself 11 times: 2(11)=2∗2∗2∗2∗2∗2∗2∗2∗2∗2∗2=2048{2^(11) = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2048} This calculation can be done manually or using a calculator. The result, 2048, represents the growth factor of the sequence up to the 12th term. It's a significant number, highlighting the exponential nature of geometric sequences.
  • Step 5: Multiply by the first term: The final step is to multiply the result from the previous step by the first term, which is 1. This step is straightforward: a12=1∗2048=2048{a_{12} = 1 * 2048 = 2048} Since multiplying by 1 doesn't change the value, the 12th term of the sequence is 2048. This final calculation brings us to the solution, completing the process of finding the 12th term of the geometric sequence.

After meticulously applying the formula and breaking down each step, we have arrived at the solution. The 12th term of the geometric sequence 1, 2, 4, 8, 16, ... is 2048. This result not only answers the specific question posed but also reinforces our understanding of geometric sequences and their properties. The process we followed can be applied to find any term in any geometric sequence, provided we know the first term and the common ratio. This answer highlights the exponential growth inherent in geometric sequences. Starting from 1, the terms quickly escalate, reaching 2048 by the 12th term. This rapid growth has implications in various real-world scenarios, such as compound interest, population growth, and the spread of information. Understanding geometric sequences allows us to model and predict these phenomena, making it a valuable tool in various fields.

In conclusion, calculating the 12th term of the geometric sequence 1, 2, 4, 8, 16, ... has been a journey through the fundamental principles of geometric sequences. We have explored the key components, applied the formula for the nth term, and broken down each step of the calculation. The final answer, 2048, stands as a testament to the power of mathematical formulas and the elegance of geometric progressions. Mastering geometric sequences is not just about memorizing formulas; it's about developing a deep understanding of the underlying concepts. This understanding allows us to apply these concepts to solve a wide range of problems, both in mathematics and in real-world applications. From financial modeling to computer science algorithms, geometric sequences play a crucial role in various fields.

This exercise in calculating the 12th term serves as a stepping stone to more advanced topics in mathematics. It lays the foundation for understanding series, limits, and other concepts that build upon the principles of sequences. By mastering these fundamentals, we equip ourselves with the tools to tackle complex problems and explore the fascinating world of mathematics. The ability to recognize patterns, apply formulas, and break down complex problems into manageable steps is a valuable skill that extends beyond the realm of mathematics. It is a skill that can be applied to problem-solving in any field, making the study of geometric sequences a worthwhile endeavor. As we continue our mathematical journey, the principles we have learned here will serve as a solid foundation for future explorations.