Finding The 7th Term In The Sequence -1, -3, -9

Understanding sequences is fundamental in mathematics, and geometric sequences, in particular, hold a significant place. These sequences exhibit a unique pattern where each term is obtained by multiplying the previous term by a constant factor, known as the common ratio. In this comprehensive guide, we will delve into the intricacies of geometric sequences, providing you with the knowledge and tools necessary to identify, analyze, and calculate specific terms within these sequences. Our focus will be on a specific sequence: -1, -3, -9, ..., and we will walk you through the process of finding its seventh term. Whether you are a student seeking to master geometric sequences or simply someone curious about mathematical patterns, this article will offer a clear and insightful exploration of this topic. The key to successfully navigating geometric sequences lies in understanding the relationship between terms and the common ratio. This foundational concept will empower you to predict future terms and solve a variety of problems related to these sequences. So, let's embark on this mathematical journey and unravel the mysteries of the sequence -1, -3, -9, ...

Identifying the Sequence as Geometric

To begin our exploration, it's crucial to establish that the given sequence, -1, -3, -9, ..., is indeed a geometric sequence. This determination hinges on the existence of a common ratio between consecutive terms. In simpler terms, we need to verify if each term is obtained by multiplying the preceding term by the same constant value. Let's examine the sequence closely. To find the ratio between the second term (-3) and the first term (-1), we divide -3 by -1, which yields 3. Now, let's check the ratio between the third term (-9) and the second term (-3). Dividing -9 by -3 also gives us 3. This consistent ratio of 3 between consecutive terms confirms that the sequence is geometric. The common ratio, denoted by 'r', is therefore 3. Identifying the common ratio is a critical step, as it forms the basis for calculating any term in the sequence. In the realm of geometric sequences, the common ratio acts as the sequence's defining characteristic, dictating its growth or decay pattern. With this fundamental understanding in place, we can proceed to develop a formula that will enable us to pinpoint the seventh term of this sequence with precision. Recognizing the geometric nature of a sequence not only allows us to predict future terms but also provides insights into the sequence's overall behavior and properties. So, having established that our sequence is geometric with a common ratio of 3, we are well-prepared to delve into the calculation of its seventh term.

Deriving the General Formula

Now that we've confirmed our sequence as geometric and identified the common ratio, our next step is to derive the general formula for a geometric sequence. This formula serves as a powerful tool, allowing us to calculate any term in the sequence without having to manually compute each preceding term. The general formula for the nth term (an) of a geometric sequence is expressed as: an = a1 * r^(n-1), where a1 represents the first term, r is the common ratio, and n denotes the term number we wish to find. This formula elegantly encapsulates the essence of geometric sequences, demonstrating how each term is a product of the initial term and the common ratio raised to a power that corresponds to the term's position in the sequence. Let's break down the formula to ensure a clear understanding of its components. The first term, a1, acts as the starting point of the sequence. The common ratio, r, dictates the multiplicative factor between consecutive terms. The exponent (n-1) reflects the fact that the first term (a1) is not multiplied by the common ratio, while the second term is multiplied by the common ratio once, the third term twice, and so on. To illustrate the formula's application, consider finding the 5th term. We would substitute n with 5, and the formula would become a5 = a1 * r^(5-1) = a1 * r^4. This highlights how the exponent adjusts to reflect the number of times the common ratio is applied to reach the desired term. With the general formula firmly in our grasp, we are now equipped to tackle the specific task of finding the seventh term in our sequence. The formula provides a systematic and efficient approach, saving us from the tedious process of repeatedly multiplying by the common ratio. In the following section, we will apply this formula to our sequence, plugging in the relevant values and obtaining the seventh term with ease.

Applying the Formula to Find the 7th Term

With the general formula for a geometric sequence firmly in place, we are now ready to apply it to our specific sequence, -1, -3, -9, ..., and find its seventh term. Recall that the formula is given by: an = a1 * r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number. In our sequence, we have already identified the first term (a1) as -1 and the common ratio (r) as 3. Our goal is to find the seventh term, which means we need to determine the value of a7. To do this, we will substitute the known values into the formula. We have a1 = -1, r = 3, and n = 7. Plugging these values into the formula, we get: a7 = -1 * 3^(7-1). Simplifying the exponent, we have: a7 = -1 * 3^6. Now, we need to calculate 3 raised to the power of 6. This is equal to 3 * 3 * 3 * 3 * 3 * 3, which results in 729. Substituting this value back into the equation, we get: a7 = -1 * 729. Finally, multiplying -1 by 729, we obtain the seventh term: a7 = -729. Therefore, the seventh term in the sequence -1, -3, -9, ... is -729. This process demonstrates the power and efficiency of the general formula. By simply substituting the relevant values, we were able to quickly and accurately determine the seventh term without having to manually calculate the preceding terms. This highlights the utility of the formula in handling geometric sequences, particularly when dealing with terms that are further along in the sequence. In the next section, we will summarize our findings and reiterate the key steps involved in finding a specific term in a geometric sequence.

Summary and Conclusion

In this exploration, we have successfully navigated the realm of geometric sequences and determined the seventh term of the sequence -1, -3, -9, .... To recap, we began by identifying the sequence as geometric, establishing the presence of a common ratio between consecutive terms. We calculated this common ratio to be 3, a crucial step in understanding the sequence's pattern. Next, we delved into the general formula for a geometric sequence, 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. This formula provides a systematic approach to finding any term in a geometric sequence. Applying this formula to our sequence, we substituted the values a1 = -1, r = 3, and n = 7. This led us to the equation a7 = -1 * 3^(7-1), which simplified to a7 = -1 * 3^6. Calculating 3^6, we obtained 729, and multiplying this by -1, we arrived at our final answer: a7 = -729. Thus, the seventh term in the sequence -1, -3, -9, ... is -729. This exercise underscores the importance of understanding the fundamental concepts of geometric sequences and the power of the general formula. By mastering these tools, you can confidently tackle a wide range of problems involving geometric sequences, efficiently calculating specific terms and gaining insights into the behavior of these mathematical patterns. Whether you are a student seeking to excel in mathematics or simply someone with a curiosity for numbers, the knowledge gained here will serve as a valuable asset in your mathematical journey. The ability to identify, analyze, and manipulate geometric sequences opens doors to a deeper understanding of mathematical relationships and their applications in various fields. As we conclude this exploration, remember that practice is key to solidifying your understanding. Continue to explore different geometric sequences, apply the formula, and challenge yourself with increasingly complex problems. With dedication and perseverance, you will master the art of working with geometric sequences and unlock a world of mathematical possibilities.