Finding The 7th Term Of The Sequence 1, -3, 9, -27, ...

This article delves into the fascinating world of geometric sequences and provides a step-by-step guide on how to determine a specific term within such a sequence. Specifically, we will focus on the sequence 1, -3, 9, -27, ... and uncover the mystery of its 7th term. Geometric sequences, characterized by a constant ratio between consecutive terms, play a crucial role in various mathematical and real-world applications. Understanding how to identify patterns and calculate specific terms within these sequences is a fundamental skill in mathematics. We will explore the underlying principles of geometric sequences, including the concept of the common ratio and the formula for finding the nth term. By applying these principles, we will systematically calculate the 7th term of the given sequence, solidifying your understanding of geometric sequences and their applications. This exploration will not only equip you with the ability to solve similar problems but also deepen your appreciation for the elegance and power of mathematical sequences.

Deciphering Geometric Sequences: The Building Blocks

Before we embark on our journey to find the 7th term, it's crucial to solidify our understanding of what constitutes a geometric sequence. A geometric sequence, at its core, is a sequence of numbers where each term is obtained by multiplying the preceding term by a constant factor. This constant factor is the lifeblood of the sequence and is aptly named the common ratio. Identifying this common ratio is paramount to unlocking the secrets of the sequence and predicting its future terms. Consider, for instance, the sequence 2, 4, 8, 16, .... Here, each term is precisely twice the preceding term. Thus, the common ratio in this sequence is 2. Similarly, in the sequence 100, 50, 25, 12.5,..., each term is half of the preceding term, making the common ratio 1/2. The common ratio can be a positive number, a negative number, or even a fraction, each imparting a unique characteristic to the geometric sequence. A positive common ratio will result in a sequence where all terms have the same sign, whereas a negative common ratio will create a sequence where the signs alternate between positive and negative. Understanding this interplay between the common ratio and the sequence's behavior is key to mastering geometric sequences. To further solidify your grasp, try identifying the common ratios in the following sequences: 3, 9, 27, 81,... and 1, -2, 4, -8, ... Remember, the common ratio is the constant multiplier that bridges one term to the next, and it's the foundation upon which we will build our understanding of geometric sequences.

Identifying the Common Ratio: The Key to Unlocking the Sequence

Now that we have a firm grasp on the definition of a geometric sequence, let's delve into the practical task of identifying the common ratio. This is the linchpin in understanding the sequence's behavior and predicting future terms. The common ratio, as we've established, is the constant factor that links each term to its predecessor. To unveil this hidden factor, we simply divide any term in the sequence by its preceding term. This seemingly simple maneuver unlocks the sequence's underlying structure. Let's illustrate this with an example. Consider the sequence 4, 12, 36, 108, .... To find the common ratio, we can divide the second term (12) by the first term (4), yielding 3. We can verify this by dividing the third term (36) by the second term (12), which also gives us 3. This consistency confirms that 3 is indeed the common ratio of this sequence. Similarly, in the sequence 16, -8, 4, -2, ..., dividing -8 by 16 gives us -1/2, and dividing 4 by -8 also gives us -1/2, solidifying -1/2 as the common ratio. It's important to note that we can choose any two consecutive terms to calculate the common ratio, as long as we consistently divide the later term by the earlier term. Mastering the art of identifying the common ratio is the cornerstone of working with geometric sequences. It empowers you to not only understand the sequence's pattern but also to predict its future terms with confidence.

The Formula for the nth Term: Predicting the Future

While we can manually calculate the terms of a geometric sequence by repeatedly multiplying by the common ratio, this method becomes cumbersome for finding terms far down the line. This is where the formula for the nth term comes to our rescue. This formula provides a direct route to calculating any term in the sequence, without having to compute all the preceding terms. The formula is elegantly expressed as: a_n = a_1 * r^(n-1), where a_n represents the nth term, a_1 is the first term, r is the common ratio, and n is the term number we're seeking. This formula encapsulates the essence of a geometric sequence, capturing the multiplicative relationship between terms and the exponential growth or decay dictated by the common ratio. Let's dissect this formula to understand its components. a_1, the first term, serves as the starting point of our sequence. r, the common ratio, is the constant multiplier that propels the sequence forward. n, the term number, specifies which term in the sequence we wish to calculate. The exponent (n-1) reflects the fact that the first term (a_1) 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. This formula is a powerful tool that allows us to bypass tedious calculations and directly access any term in a geometric sequence. In the next section, we will apply this formula to find the elusive 7th term of our sequence.

Finding the 7th Term: Putting the Formula to Work

Now, armed with the formula for the nth term, we can confidently embark on our quest to find the 7th term of the sequence 1, -3, 9, -27, .... Let's revisit the formula: a_n = a_1 * r^(n-1). To apply this formula, we need to identify the values of a_1, r, and n for our specific sequence. The first term, a_1, is clearly 1. To find the common ratio, r, we can divide any term by its preceding term. For instance, dividing -3 by 1 gives us -3, and dividing 9 by -3 also yields -3. Thus, the common ratio, r, is -3. We are seeking the 7th term, so n is 7. Now, we have all the necessary ingredients to plug into our formula. Substituting the values, we get: a_7 = 1 * (-3)^(7-1). Simplifying the exponent, we have: a_7 = 1 * (-3)^6. Now, we need to calculate (-3)^6. Remember that a negative number raised to an even power results in a positive number. (-3)^6 = (-3) * (-3) * (-3) * (-3) * (-3) * (-3) = 729. Therefore, a_7 = 1 * 729 = 729. Thus, the 7th term of the sequence 1, -3, 9, -27, ... is 729. This methodical application of the formula demonstrates the power of mathematical tools in unraveling the patterns hidden within sequences. We have successfully navigated the sequence and pinpointed its 7th term, showcasing the elegance and efficiency of the nth term formula.

Conclusion: Mastering Geometric Sequences

In this comprehensive exploration, we have successfully unveiled the 7th term of the geometric sequence 1, -3, 9, -27, ..., demonstrating a thorough understanding of geometric sequences and the power of the nth term formula. We began by defining geometric sequences and emphasizing the critical role of the common ratio. We then honed our skills in identifying the common ratio, a fundamental step in deciphering the sequence's behavior. The introduction of the formula for the nth term provided us with a powerful tool for predicting future terms, bypassing the need for tedious manual calculations. Applying this formula, we systematically calculated the 7th term, solidifying our understanding of its practical application. This journey through geometric sequences not only equips you with the ability to solve similar problems but also fosters a deeper appreciation for the elegance and efficiency of mathematical tools. Mastering geometric sequences opens doors to a wider understanding of mathematical patterns and their applications in various fields, from finance and computer science to physics and biology. Embrace the power of these sequences, and you'll find yourself equipped to tackle a wide array of mathematical challenges.