Geometric Sequences Finding The Nth Term

Geometric sequences are a fundamental concept in mathematics, and understanding how to work with them is crucial for various applications. In this article, we'll dive deep into geometric sequences and tackle the problem of finding a specific term within a given sequence. We will focus on the series -128, 64, -32, ..., and systematically determine its 12th term. To effectively navigate geometric sequences, it's vital to grasp their core components, including the first term, the common ratio, and the general formula for finding any term in the sequence. Let's first define what a geometric sequence is and then delve into the specifics of solving this problem.

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant value, known as the common ratio (r). Identifying the common ratio is the first step in solving problems related to geometric sequences. In our given series, -128, 64, -32, ..., we can find the common ratio by dividing any term by its preceding term. For instance, dividing 64 by -128 gives us -1/2, and dividing -32 by 64 also yields -1/2. This consistency confirms that our common ratio (r) is -1/2. Once we have the common ratio, we can use the general formula for the nth term of a geometric sequence. The formula is expressed as an = a1 * r^(n-1), where an represents the nth term, a1 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 12th term (n = 12), the first term (a1) is -128, and the common ratio (r) is -1/2.

Applying the formula to find the 12th term (a12), we substitute the values: a12 = -128 * (-1/2)^(12-1). This simplifies to a12 = -128 * (-1/2)^11. Calculating (-1/2)^11 gives us -1/2048. Now, we multiply -128 by -1/2048: a12 = -128 * (-1/2048) = 128/2048. Simplifying this fraction, we divide both the numerator and denominator by 128, resulting in 1/16. Therefore, the 12th term of the series -128, 64, -32, ... is 1/16. Understanding and applying the formula correctly is essential to solving these types of problems accurately. The process involves identifying the key components of the sequence, such as the first term and the common ratio, and then using these values in the general formula. This step-by-step approach ensures clarity and precision in finding the desired term. In conclusion, by carefully analyzing the geometric sequence and applying the appropriate formula, we have successfully determined that the 12th term of the given series is 1/16, which corresponds to option (c).

Answer: c) 1/16

Find the 4th Term of the Series 0.04, 0.2, 1, ...

Continuing our exploration of geometric sequences, let's address the task of finding a specific term in another series. This time, we're focusing on the series 0.04, 0.2, 1, ... and our goal is to determine its 4th term. Similar to our previous problem, the key to solving this lies in understanding the underlying principles of geometric sequences and applying the appropriate formula. Recall that a geometric sequence is characterized by a common ratio (r), which is the constant value by which each term is multiplied to obtain the next term. To find this common ratio, we divide any term by its preceding term. This process helps us reveal the pattern within the sequence and sets the stage for finding any term we desire. In this specific series, the initial terms provide us with the information needed to calculate the common ratio and subsequently determine the 4th term.

In the series 0.04, 0.2, 1, ..., let's find the common ratio (r). Dividing 0.2 by 0.04, we get 5. Similarly, dividing 1 by 0.2 also gives us 5. This consistency confirms that the common ratio (r) is indeed 5. Now that we have the common ratio, we can proceed to find the 4th term. The formula for the nth term of a geometric sequence is an = a1 * r^(n-1), where an represents the nth term, a1 is the first term, r is the common ratio, and n is the term number we want to find. In this case, we want to find the 4th term (n = 4), the first term (a1) is 0.04, and the common ratio (r) is 5. Substituting these values into the formula, we get a4 = 0.04 * 5^(4-1). This simplifies to a4 = 0.04 * 5^3. Calculating 5^3 gives us 125. Now, we multiply 0.04 by 125: a4 = 0.04 * 125 = 5. Thus, the 4th term of the series 0.04, 0.2, 1, ... is 5.

This methodical approach demonstrates how understanding the properties of geometric sequences allows us to efficiently find any term in the sequence. By calculating the common ratio and applying the general formula, we can accurately determine the value of the 4th term. This process involves careful attention to detail and a clear understanding of the formula's components. The ability to identify patterns and apply formulas is crucial in solving mathematical problems, especially those involving sequences and series. Therefore, in the context of the given series, 0.04, 0.2, 1, ..., we have successfully found that the 4th term is 5, which corresponds to option (c). This exercise highlights the importance of understanding the fundamental principles of geometric sequences and applying them effectively. In conclusion, by systematically analyzing the sequence and utilizing the formula for the nth term, we accurately determined the 4th term to be 5, reinforcing the importance of a structured approach to problem-solving in mathematics. Understanding the steps involved in solving geometric sequences will help you tackle more complex problems in the future.

Answer: c) 5

Find the Last Term of the Series 1, 2, 4, ... to 10 Terms

Let's tackle another intriguing problem related to geometric sequences: finding the last term of a series. This time, we're presented with the series 1, 2, 4, ... and our objective is to determine the 10th term. This problem further reinforces the importance of understanding the core principles of geometric sequences and applying the appropriate formulas to arrive at the correct solution. As before, the first step in solving this problem is to identify the common ratio (r). This ratio is the constant value by which each term is multiplied to obtain the subsequent term. Identifying this ratio is crucial because it allows us to predict and calculate any term in the sequence. Once we've determined the common ratio, we can utilize the general formula for the nth term of a geometric sequence to find the 10th term.

In the series 1, 2, 4, ..., the common ratio (r) can be found by dividing any term by its preceding term. For example, dividing 2 by 1 gives us 2, and dividing 4 by 2 also yields 2. This confirms that the common ratio (r) is 2. With the common ratio determined, we can now find the 10th term. The formula for the nth term of a geometric sequence is an = a1 * r^(n-1), where an represents the nth term, a1 is the first term, r is the common ratio, and n is the term number we want to find. In this case, we want to find the 10th term (n = 10), the first term (a1) is 1, and the common ratio (r) is 2. Substituting these values into the formula, we get a10 = 1 * 2^(10-1). This simplifies to a10 = 1 * 2^9. Calculating 2^9 gives us 512. Therefore, the 10th term of the series 1, 2, 4, ... is 512.

This problem underscores the importance of a clear and structured approach when dealing with mathematical sequences. By identifying the common ratio and applying the general formula, we can efficiently and accurately determine the value of any term in the sequence. The process involves understanding the relationship between the terms and utilizing the formula to extrapolate to the desired term. The ability to work with geometric sequences is a valuable skill in mathematics, with applications in various fields. By systematically analyzing the series and applying the appropriate formula, we successfully found that the 10th term of the given sequence is 512. This exercise emphasizes the importance of a methodical approach to problem-solving and the application of fundamental mathematical principles. In conclusion, by carefully following the steps and using the formula for the nth term, we accurately determined the last term (10th term) of the given series to be 512, highlighting the practical application of geometric sequence principles in mathematical problem-solving.

Answer: 512