Finding The 6th Term In The Sequence -1, 4, -16, 64

July 13, 2025 by Scholario Team 52 views

Finding the 6th Term in a Geometric Sequence

Are you grappling with geometric sequences and need to pinpoint a specific term? In this comprehensive guide, we'll dissect the given sequence $-1, 4, -16, 64, \ldots$ and master the techniques to efficiently determine the $6^{\text{th}}$ term. Whether you're a student tackling math problems or simply intrigued by the elegance of mathematical patterns, this step-by-step explanation will equip you with the knowledge and confidence to solve similar problems. Let's embark on this mathematical journey together!

Understanding Geometric Sequences

Before we dive into the specifics of this particular sequence, let's first solidify our understanding of geometric sequences in general. A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a constant value. This constant value is called the common ratio, often denoted by the variable 'r'. For example, the sequence 2, 6, 18, 54,... is a geometric sequence because each term is three times the previous term. Here, the common ratio is 3. Recognizing a geometric sequence is the first step in finding any specific term within it. The beauty of geometric sequences lies in their predictable pattern, which allows us to develop formulas and methods for calculating any term, no matter how far down the sequence it lies.

The common ratio is the cornerstone of any geometric sequence. Finding it allows us to understand the relationship between consecutive terms and ultimately predict future terms. To calculate the common ratio (r), simply divide any term by its preceding term. It's crucial to consistently use the same terms to avoid errors. For instance, in the sequence 2, 6, 18, the common ratio can be found by dividing 6 by 2 (which equals 3) or dividing 18 by 6 (which also equals 3). Understanding this fundamental concept is key to working with geometric sequences. Once we have the common ratio, we can use it to construct the general formula for the sequence and calculate any term we desire.

Now, let’s explore the general formula for a geometric sequence, a powerful tool that unlocks the ability to find any term without listing out the entire sequence. The general formula is expressed as: $a_n = a_1 * r^(n-1)$ where:

$a_n$ represents the nth term in the sequence (the term we want to find).
$a_1$ represents the first term of the sequence.
r represents the common ratio.
n represents the position of the term we want to find (e.g., for the 6th term, n = 6).

This formula encapsulates the essence of a geometric sequence, highlighting how each term is derived from the first term and the common ratio. By substituting the known values into this formula, we can directly calculate the value of any term, saving us the time and effort of manually extending the sequence. The formula allows us to leapfrog through the sequence, jumping directly to the term of interest. Mastering the use of this formula is essential for efficiently solving problems involving geometric sequences.

Analyzing the Given Sequence: -1, 4, -16, 64, ...

Let's shift our focus to the specific sequence at hand: $-1, 4, -16, 64, \ldots$ . The first step in deciphering this sequence is to confirm that it is indeed geometric. To do this, we will calculate the ratio between consecutive terms. If the ratio remains constant, we can confidently classify the sequence as geometric.

To determine the common ratio (r), we'll divide the second term by the first term, the third term by the second term, and so on. This process will reveal the multiplicative factor that defines the sequence. Dividing 4 by -1 yields -4. Dividing -16 by 4 also yields -4. Finally, dividing 64 by -16 results in -4 as well. The consistent ratio of -4 confirms that the sequence is indeed geometric, and our common ratio (r) is -4. Identifying the common ratio is a crucial step, as it forms the basis for calculating any term in the sequence. This negative common ratio also indicates that the terms will alternate in sign, a characteristic we can already observe in the given sequence.

Now that we have established the sequence as geometric and identified the common ratio, let's pinpoint the first term ( $a_1$ ). In the given sequence, the first term is clearly -1. This value, along with the common ratio, will be the foundation for calculating any other term in the sequence. Recognizing the first term is often a straightforward task, but it's essential to explicitly state it as it's a key component in the general formula. With both the first term and the common ratio in hand, we are well-equipped to utilize the general formula and find the 6th term.

Calculating the 6th Term

Now we arrive at the crux of the problem: finding the 6th term. We will leverage the general formula for a geometric sequence, $a_n = a_1 * r^(n-1)$ , and substitute the values we've identified: $a_1 = -1$ , $r = -4$ , and $n = 6$ (since we are looking for the 6th term). This substitution will allow us to directly calculate the value of the 6th term without having to manually extend the sequence.

Let's substitute the known values into the general formula: $a_6 = (-1) * (-4)^(6-1)$ . This simplifies to $a_6 = (-1) * (-4)^5$ . Now, we need to calculate $(-4)^5$ , which means multiplying -4 by itself five times. $(-4)^5 = -1024$ . This negative result is due to the odd exponent applied to a negative base. Finally, we multiply -1 by -1024: $a_6 = (-1) * (-1024) = 1024$ . Therefore, the 6th term in the sequence is 1024. We have successfully used the general formula to efficiently determine the 6th term.

Verification and Conclusion

To ensure the accuracy of our result, we can manually extend the sequence to the 6th term. We have the first four terms: -1, 4, -16, 64. To find the 5th term, we multiply the 4th term (64) by the common ratio (-4): $64 * (-4) = -256$ . To find the 6th term, we multiply the 5th term (-256) by the common ratio (-4): $-256 * (-4) = 1024$ . This manual calculation confirms our previous result obtained using the general formula. Manual verification, although time-consuming, provides an extra layer of confidence in our solution. It also reinforces the understanding of how each term in a geometric sequence is generated.

In conclusion, the 6th term in the sequence $-1, 4, -16, 64, \ldots$ is 1024. We successfully determined this by first recognizing the sequence as geometric, calculating the common ratio, identifying the first term, and then applying the general formula for a geometric sequence. This problem highlights the power and efficiency of using mathematical formulas to solve problems. Understanding and applying these fundamental concepts allows us to tackle complex sequences with ease and precision. By mastering these techniques, you'll be well-equipped to solve a wide range of problems involving geometric sequences.

Practice Problems

To further solidify your understanding, try solving these practice problems:

Find the 8th term in the sequence 3, 6, 12, 24, ...
Find the 5th term in the sequence 100, -50, 25, -12.5, ...
Find the 7th term in the sequence 2, -6, 18, -54, ...

By working through these examples, you'll gain further confidence in applying the concepts and techniques discussed in this guide. Remember to identify the common ratio and the first term before applying the general formula. Good luck, and happy problem-solving!