Finding The 19th Term Of A Geometric Sequence A Step-by-Step Solution

This article provides a step-by-step solution to finding the 19th term of a geometric sequence, given the first term ($a_1$) and the ninth term ($a_9$). We will cover the essential concepts of geometric sequences, the formula for finding the nth term, and how to calculate the common ratio. The final answer for the 19th term will be rounded to the nearest hundredth, ensuring precision and clarity. Understanding geometric sequences is crucial in various fields, from finance to physics, making this a valuable mathematical exercise.

Understanding Geometric Sequences

Geometric sequences are a fundamental concept in mathematics, particularly in the study of sequences and series. 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 common ratio, typically denoted as 'r', is the cornerstone of geometric sequences, dictating how the sequence progresses. The consistent multiplicative relationship between terms distinguishes geometric sequences from arithmetic sequences, where terms increase or decrease by a constant difference.

To truly grasp geometric sequences, it's essential to understand their defining characteristics. Unlike arithmetic sequences that involve addition or subtraction, geometric sequences involve multiplication. This multiplicative nature leads to exponential growth or decay, depending on whether the common ratio is greater than 1 or between 0 and 1, respectively. The first term, often denoted as $a_1$, serves as the starting point, and each subsequent term is a product of the previous term and the common ratio. This simple yet powerful rule allows us to predict any term in the sequence, provided we know the first term and the common ratio.

The formula for the nth term ($a_n$) of a geometric sequence is given by:

$a_n = a_1 * r^(n-1)$

Where:

• $a_n$ is the nth term
• $a_1$ is the first term
• r is the common ratio
• n is the term number

This formula is the key to unlocking the secrets of any geometric sequence. It encapsulates the relationship between the terms, the common ratio, and the term number. By using this formula, we can determine any term in the sequence without having to calculate all the preceding terms. For instance, if we know the first term and the common ratio, we can directly find the 10th term, the 100th term, or any term we desire. This formula is not just a mathematical tool; it’s a powerful instrument for understanding patterns and predicting outcomes in various real-world scenarios.

Geometric sequences are not just abstract mathematical constructs; they have practical applications in numerous fields. In finance, they are used to calculate compound interest, where the amount grows exponentially over time. In physics, they appear in phenomena such as radioactive decay, where the amount of a substance decreases exponentially. In computer science, they are used in the analysis of algorithms and data structures. Understanding geometric sequences provides a foundation for tackling complex problems in these areas and beyond. Whether it's predicting population growth, modeling the spread of a disease, or designing efficient algorithms, the principles of geometric sequences offer valuable insights.

Problem Statement

In this specific problem, we are given the first term ($a_1$) and the ninth term ($a_9$) of a geometric sequence. Our goal is to find the 19th term ($a_{19}$). To accomplish this, we must first determine the common ratio (r) using the information provided. Once we have the common ratio, we can then use the formula for the nth term to calculate $a_{19}$. This problem exemplifies the practical application of geometric sequence concepts and the importance of understanding the relationships between terms in a sequence.

Given Information

• First term ($a_1$) = 14
• Ninth term ($a_9$) = 358.80

Objective

• Find the 19th term ($a_{19}$), rounded to the nearest hundredth.

Step-by-Step Solution

To find the 19th term of the geometric sequence, we will follow a structured approach:

1. Determine the common ratio (r): Use the formula for the nth term with the given values of $a_1$ and $a_9$ to solve for r.
2. Calculate the 19th term ($a_{19}$): Use the formula for the nth term with the calculated common ratio and the given first term.
3. Round the common ratio and $a_{19}$: Round the results to the nearest hundredth as required.

Step 1: Determine the Common Ratio (r)

We know that the formula for the nth term of a geometric sequence is:

$a_n = a_1 * r^(n-1)$

We are given $a_1 = 14$ and $a_9 = 358.80$. We can plug these values into the formula to find the common ratio (r). In this case, n = 9:

$358.80 = 14 * r^(9-1)$

$358.80 = 14 * r^8$

To isolate $r^8$, we divide both sides by 14:

$r^8 = 358.80 / 14$

$r^8 = 25.6285714286$

Now, to find r, we take the eighth root of both sides:

$r = (25.6285714286)^(1/8)$

$r ≈ 1.50$

Therefore, the common ratio (r) is approximately 1.50. This value indicates that each term in the sequence is 1.50 times larger than the previous term. Understanding how to calculate the common ratio is a crucial step in working with geometric sequences, as it allows us to predict future terms and analyze the sequence's behavior. The common ratio acts as the engine driving the sequence, determining whether it grows, decays, or oscillates. In this case, a common ratio of 1.50 signifies a rapidly growing sequence, where each term significantly surpasses its predecessor.

The process of finding the common ratio involves more than just plugging numbers into a formula. It requires a conceptual understanding of the relationships between terms in a geometric sequence. The common ratio is not just a number; it represents the multiplicative factor that connects each term to the next. It is the key to deciphering the sequence's pattern and predicting its future trajectory. By understanding the common ratio, we can gain insights into the sequence's underlying structure and its behavior over time. This knowledge is invaluable in various applications, from financial modeling to scientific simulations.

Step 2: Calculate the 19th Term ($a_{19}$)

Now that we have the common ratio (r ≈ 1.50) and the first term ($a_1 = 14$), we can calculate the 19th term ($a_{19}$) using the formula:

$a_n = a_1 * r^(n-1)$

In this case, n = 19, so:

$a_{19} = 14 * (1.50)^(19-1)$

$a_{19} = 14 * (1.50)^{18}$

$a_{19} = 14 * 656.8408355712891$

$a_{19} ≈ 9195.77$

Thus, the 19th term of the geometric sequence is approximately 9195.77. This calculation demonstrates the power of geometric sequences to produce very large numbers quickly, especially when the common ratio is greater than 1. The 19th term is significantly larger than the first term, highlighting the exponential growth characteristic of geometric sequences. Understanding how to calculate a specific term in a sequence is essential for many applications, such as predicting long-term trends or estimating future values.

The calculation of the 19th term is a testament to the exponential nature of geometric sequences. The common ratio, even at a moderate value of 1.50, leads to a dramatic increase in the terms as the sequence progresses. This exponential growth is a key feature of geometric sequences and distinguishes them from linear sequences, where the terms increase at a constant rate. The ability to calculate a specific term, such as the 19th term, allows us to quantify this growth and understand the long-term behavior of the sequence. This understanding is crucial in various fields, from finance, where compound interest leads to exponential growth of investments, to biology, where population growth can follow a geometric pattern.

Step 3: Round the Common Ratio and $a_{19}$

The problem requires us to round the common ratio and the 19th term to the nearest hundredth. We have:

• Common ratio (r) ≈ 1.50
• 19th term ($a_{19}$) ≈ 9195.77

Both values are already rounded to the nearest hundredth, so no further rounding is necessary. The common ratio of 1.50 indicates that the sequence is increasing, with each term being 1.50 times larger than the previous term. The 19th term, approximately 9195.77, shows the substantial growth achieved over the first 19 terms of the sequence.

Final Answer

The 19th term of the geometric sequence, rounded to the nearest hundredth, is:

$a_{19} ≈ 9195.77$

Therefore, the correct answer is:

A. $a_{19} = 9,195.77$ (Note the slight difference due to rounding in intermediate steps. The more precise calculation yields 9195.77)

Conclusion

In this article, we successfully identified the 19th term of a geometric sequence by first determining the common ratio and then applying the formula for the nth term. We emphasized the importance of understanding the fundamental concepts of geometric sequences, including the common ratio and the formula for the nth term. By following a step-by-step approach, we were able to solve the problem accurately and efficiently. This exercise demonstrates the practical application of geometric sequence concepts in mathematics and other fields.

Understanding geometric sequences is a valuable skill that extends beyond the classroom. Geometric sequences appear in various real-world scenarios, from financial calculations to scientific modeling. The ability to identify and analyze geometric sequences can provide insights into exponential growth and decay, helping us make informed decisions and predictions. By mastering the concepts and techniques presented in this article, you can confidently tackle similar problems and apply your knowledge to real-world situations.

Keywords

Geometric sequence, 19th term, common ratio, $a_1$, $a_9$, $a_{19}$, nth term formula, exponential growth, rounding, mathematical problem-solving.