Finding The 6th Term In A Geometric Sequence Explained

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In the realm of mathematics, geometric sequences hold a prominent position, showcasing a captivating pattern of numbers that unfold with a constant multiplicative factor. Understanding these sequences is crucial for various applications, from financial calculations to modeling natural phenomena. In this comprehensive guide, we will delve into the intricacies of geometric sequences, focusing on how to determine the nth term using an explicit formula. Specifically, we will address the question of finding the 6th term in a geometric sequence defined by the explicit formula: an=−5imes(4)(n−1)a_n = -5 imes (4)^{(n-1)}. By the end of this exploration, you will be equipped with the knowledge and skills to confidently tackle similar problems and appreciate the elegance of geometric sequences.

Deciphering Geometric Sequences: A Foundation for Understanding

Before we embark on the journey of finding the 6th term, let's first establish a firm understanding of what geometric sequences are. A geometric sequence is a sequence of numbers where each term is obtained by multiplying the preceding term by a constant factor, known as the common ratio. This constant ratio is the linchpin that governs the progression of the sequence, dictating how each term relates to its predecessor. The beauty of geometric sequences lies in their predictable nature, allowing us to extrapolate patterns and determine any term in the sequence with relative ease.

To illustrate this concept, consider the sequence 2, 6, 18, 54, and so on. Here, the common ratio is 3, as each term is obtained by multiplying the previous term by 3. This multiplicative relationship defines the geometric nature of the sequence. In contrast, an arithmetic sequence progresses by adding a constant difference between terms, rather than multiplying. The distinction between these two types of sequences is fundamental in mathematics.

Understanding the concept of a common ratio is paramount when working with geometric sequences. The common ratio, often denoted as 'r', is the constant factor that links consecutive terms. It can be a positive or negative number, an integer, or a fraction. The common ratio determines whether the sequence is increasing (when r > 1), decreasing (when 0 < r < 1), or alternating (when r < 0). Identifying the common ratio is often the first step in analyzing a geometric sequence.

Moreover, the first term of the sequence, often denoted as 'a' or 'a1a_1', serves as the starting point for the sequence's progression. It is the initial value from which all subsequent terms are generated. Together, the first term and the common ratio provide a complete blueprint for constructing the entire geometric sequence. With these two pieces of information, we can determine any term in the sequence, regardless of its position.

Unveiling the Explicit Formula: A Gateway to Finding Any Term

The explicit formula is a powerful tool that allows us to directly calculate any term in a geometric sequence without having to compute all the preceding terms. This formula encapsulates the essence of the sequence's pattern, providing a concise and efficient way to determine any term based on its position. The explicit formula for a geometric sequence is generally expressed as:

an=a1imesr(n−1)a_n = a_1 imes r^{(n-1)}

where:

  • ana_n represents the nth term of the sequence
  • a1a_1 is the first term of the sequence
  • r is the common ratio
  • n is the position of the term in the sequence (i.e., 1st, 2nd, 3rd, etc.)

This formula elegantly captures the multiplicative nature of geometric sequences. The term r(n−1)r^{(n-1)} signifies that the common ratio is raised to the power of (n-1), which reflects the number of times the common ratio is multiplied to reach the nth term from the first term. The first term, a1a_1, acts as the initial value that is scaled by this multiplicative factor.

The explicit formula is invaluable because it provides a direct link between the term number (n) and the value of the term (ana_n). This eliminates the need to recursively calculate terms, which can be time-consuming for large values of n. With the explicit formula, we can jump directly to the term we seek, making it an indispensable tool for analyzing geometric sequences.

Decoding the Given Explicit Formula: Setting the Stage for Calculation

In our specific problem, we are presented with the explicit formula:

an=−5imes(4)(n−1)a_n = -5 imes (4)^{(n-1)}

This formula defines a particular geometric sequence, and our task is to find the 6th term. To effectively utilize this formula, we must first identify the key components: the first term (a1a_1) and the common ratio (r). By carefully examining the formula, we can deduce these values.

Comparing the given formula to the general form of the explicit formula, we can see that the first term, a1a_1, corresponds to the coefficient multiplying the exponential term. In this case, a1a_1 is -5. This negative value indicates that the sequence will have alternating signs, as each term will be a multiple of -5.

The common ratio, r, is the base of the exponential term. In our formula, the base is 4. This signifies that each term in the sequence is four times the previous term. Since the common ratio is greater than 1, we can expect the sequence to increase in magnitude as we progress through the terms.

With the first term and the common ratio identified, we have all the necessary ingredients to calculate any term in the sequence. The stage is now set for us to find the 6th term, which is the ultimate goal of our exploration.

Finding the 6th Term: Applying the Explicit Formula

Now that we have deciphered the explicit formula and identified the first term (a1=−5a_1 = -5) and the common ratio (r = 4), we can proceed to calculate the 6th term (a6a_6). To do this, we simply substitute n = 6 into the explicit formula:

a6=−5imes(4)(6−1)a_6 = -5 imes (4)^{(6-1)}

Simplifying the exponent, we get:

a6=−5imes(4)5a_6 = -5 imes (4)^5

Now, we need to calculate 4 raised to the power of 5. This can be done either manually or using a calculator. 4 to the power of 5 is 4 * 4 * 4 * 4 * 4, which equals 1024. Substituting this value back into the equation, we get:

a6=−5imes1024a_6 = -5 imes 1024

Finally, multiplying -5 by 1024, we obtain:

a6=−5120a_6 = -5120

Therefore, the 6th term in the geometric sequence defined by the explicit formula an=−5imes(4)(n−1)a_n = -5 imes (4)^{(n-1)} is -5120. This result aligns with the options provided, and we can confidently select the correct answer.

Verifying the Result: Ensuring Accuracy and Understanding

To ensure the accuracy of our calculation and further solidify our understanding, it's always a good practice to verify the result. One way to do this is to calculate the first few terms of the sequence and observe the pattern. This can help us confirm that our calculated 6th term fits within the sequence's progression.

Let's calculate the first few terms:

  • a1=−5imes(4)(1−1)=−5imes(4)0=−5imes1=−5a_1 = -5 imes (4)^{(1-1)} = -5 imes (4)^0 = -5 imes 1 = -5
  • a2=−5imes(4)(2−1)=−5imes(4)1=−5imes4=−20a_2 = -5 imes (4)^{(2-1)} = -5 imes (4)^1 = -5 imes 4 = -20
  • a3=−5imes(4)(3−1)=−5imes(4)2=−5imes16=−80a_3 = -5 imes (4)^{(3-1)} = -5 imes (4)^2 = -5 imes 16 = -80
  • a4=−5imes(4)(4−1)=−5imes(4)3=−5imes64=−320a_4 = -5 imes (4)^{(4-1)} = -5 imes (4)^3 = -5 imes 64 = -320
  • a5=−5imes(4)(5−1)=−5imes(4)4=−5imes256=−1280a_5 = -5 imes (4)^{(5-1)} = -5 imes (4)^4 = -5 imes 256 = -1280

We can see that the sequence is progressing as expected, with each term being four times the previous term and alternating in sign. The 6th term, -5120, logically follows this pattern, further validating our calculation.

Another way to verify the result is to use a calculator or a computer program to generate the sequence and check the 6th term. This provides an independent confirmation of our answer.

Conclusion: Mastering Geometric Sequences and the Explicit Formula

In this comprehensive guide, we have successfully navigated the world of geometric sequences and conquered the challenge of finding the 6th term using an explicit formula. We began by laying a solid foundation of understanding geometric sequences, emphasizing the crucial roles of the common ratio and the first term. We then delved into the power of the explicit formula, showcasing its ability to directly calculate any term in a sequence.

Applying the explicit formula to our specific problem, we meticulously calculated the 6th term of the geometric sequence defined by an=−5imes(4)(n−1)a_n = -5 imes (4)^{(n-1)}, arriving at the answer of -5120. We further reinforced our confidence by verifying the result through pattern observation and alternative calculation methods.

By mastering the concepts and techniques presented in this guide, you are now well-equipped to tackle a wide range of problems involving geometric sequences. Whether you are determining specific terms, analyzing patterns, or applying these sequences to real-world scenarios, the knowledge you have gained will serve you well. Geometric sequences are a fundamental building block in mathematics, and your understanding of them will undoubtedly enhance your mathematical prowess.

Remember, the key to success in mathematics lies in understanding the underlying principles and practicing consistently. So, continue to explore the fascinating world of sequences and series, and you will undoubtedly achieve greater mathematical heights.

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Finding the 6th Term in a Geometric Sequence Explained

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What is the 6th term in the geometric sequence given the explicit formula an=−5imes(4)(n−1)a_n=-5 imes (4)^{(n-1)}?