Finding The 6th Term In The Sequence 5, 20, 80, ...

by ADMIN 52 views

Finding a specific term in a sequence can be straightforward if you know the underlying pattern. In mathematics, sequences are ordered lists of numbers, and one common type is a geometric sequence. In this article, we will delve into how to find the 6th term of the geometric sequence 5, 20, 80, ... using a specific formula. We will also discuss the key concepts behind geometric sequences and common ratios to provide a comprehensive understanding of the topic.

Understanding Geometric Sequences

To find the 6th term, we must first understand what a geometric sequence is. A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a constant value, called the common ratio. This constant ratio is the key characteristic that defines a geometric sequence. Recognizing this pattern is crucial for predicting future terms in the sequence.

In the given sequence, 5, 20, 80, ..., we observe that each term is multiplied by a certain number to get the next term. To identify this common ratio, divide any term by its preceding term. For instance, dividing 20 by 5 gives us 4, and dividing 80 by 20 also yields 4. Thus, the common ratio in this sequence is 4. This understanding is the foundation for calculating any term in the sequence, including the 6th term.

Identifying the common ratio is pivotal in working with geometric sequences. Once the common ratio is determined, we can apply it to predict subsequent terms in the sequence. This predictive capability is particularly useful in various mathematical and real-world applications, such as financial growth calculations, population modeling, and compound interest calculations. The ability to extrapolate future terms from a geometric sequence based on its common ratio makes it a powerful tool in quantitative analysis.

The Formula for the nth Term

To find the 6th term, or any term in a geometric sequence, we use a specific formula. This formula streamlines the process and allows us to calculate any term without having to list out all the preceding terms. The formula for the nth term (an{a_n}) of a geometric sequence is given by:

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

Where:

  • an{ a_n } is the nth term we want to find.
  • a1{ a_1 } is the first term of the sequence.
  • r is the common ratio.
  • n is the term number we are looking for.

This formula is derived from the fundamental principle of geometric sequences: each term is the product of the first term and the common ratio raised to a power that corresponds to the term's position in the sequence. The exponent (n - 1) arises because the first term (a1{a_1}) is not multiplied by the common ratio, the second term is multiplied by the common ratio once, the third term twice, and so on. Therefore, the nth term is multiplied by the common ratio (n - 1) times.

Applying this formula allows for the efficient computation of terms far down the sequence. Rather than manually multiplying the common ratio successively, this formula provides a direct route to the desired term. It is a crucial tool in dealing with geometric sequences and series, and mastering its application is essential for tackling related mathematical problems.

Applying the Formula to Find the 6th Term

Now, to find the 6th term of the sequence 5, 20, 80, ..., we can directly apply the formula discussed above. We have all the necessary components: the first term (a1{a_1}), the common ratio (r), and the term number (n).

From the sequence, we know that the first term (a1{a_1}) is 5. We have already determined that the common ratio (r) is 4. Since we want to find the 6th term, n is 6. Plugging these values into the formula, we get:

a6=5imes4(6−1){ a_6 = 5 imes 4^{(6-1)} }

Simplifying the exponent, we have:

a6=5imes45{ a_6 = 5 imes 4^5 }

Now, we calculate 4 to the power of 5:

45=4imes4imes4imes4imes4=1024{ 4^5 = 4 imes 4 imes 4 imes 4 imes 4 = 1024 }

Finally, we multiply this result by 5:

a6=5imes1024=5120{ a_6 = 5 imes 1024 = 5120 }

Therefore, the 6th term of the sequence is 5120. This step-by-step calculation illustrates how the formula efficiently provides the desired term. By identifying the first term, the common ratio, and the term number, we can accurately determine any term in the geometric sequence without manually calculating all preceding terms. This application of the formula demonstrates its practical utility in solving sequence-related problems.

Step-by-Step Solution

To find the 6th term, let's break down the solution step-by-step for clarity:

  1. Identify the First Term (a1{a_1}): The first term of the sequence is 5.

  2. Determine the Common Ratio (r): Divide any term by its preceding term. For example, 20 / 5 = 4, and 80 / 20 = 4. So, the common ratio is 4.

  3. Identify the Term Number (n): We want to find the 6th term, so n = 6.

  4. Apply the Formula: Use the formula for the nth term of a geometric sequence:

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

    Substitute the values:

    a6=5imes4(6−1){ a_6 = 5 imes 4^{(6-1)} }

  5. Simplify the Exponent:

    a6=5imes45{ a_6 = 5 imes 4^5 }

  6. Calculate 45{4^5}:

    45=1024{ 4^5 = 1024 }

  7. Multiply:

    a6=5imes1024{ a_6 = 5 imes 1024 }

    a6=5120{ a_6 = 5120 }

Thus, the 6th term of the geometric sequence 5, 20, 80, ... is 5120. This step-by-step approach ensures clarity and accuracy in the calculation. Each step logically builds upon the previous one, making it easy to follow and understand the process. This methodical approach is essential in solving mathematical problems, as it reduces the likelihood of errors and provides a clear pathway to the solution.

Importance of Common Ratio

The common ratio is a critical element in geometric sequences. It dictates the rate at which the sequence grows or diminishes. Understanding and correctly identifying the common ratio is paramount for accurately predicting future terms in the sequence.

In the sequence 5, 20, 80, ..., the common ratio of 4 indicates that each term is four times the previous term. This multiplicative relationship is what defines the geometric progression. A common ratio greater than 1, as in this case, results in an increasing sequence, where each term is larger than the preceding one. Conversely, a common ratio between 0 and 1 leads to a decreasing sequence, where each term is smaller than the previous one.

The significance of the common ratio extends beyond simple sequence prediction. It is a fundamental parameter in various mathematical and real-world applications. For instance, in finance, the common ratio is used to calculate compound interest, where the interest earned in each period is reinvested, leading to exponential growth. In population dynamics, the common ratio can model the growth rate of a population over time. Similarly, in physics, it can describe processes involving exponential decay, such as radioactive decay.

Furthermore, the common ratio is crucial in determining the convergence or divergence of geometric series. A geometric series is the sum of the terms in a geometric sequence. If the absolute value of the common ratio is less than 1, the series converges to a finite sum; otherwise, it diverges. This concept is essential in calculus and mathematical analysis. Thus, the common ratio's importance cannot be overstated, as it underpins many theoretical and practical applications across various disciplines.

Conclusion

In conclusion, to find the 6th term of the geometric sequence 5, 20, 80, ..., we utilized the formula an=a1×r(n−1){ a_n = a_1 \times r^{(n-1)} }. By identifying the first term as 5, the common ratio as 4, and the term number as 6, we calculated the 6th term to be 5120. This process illustrates the power and efficiency of using the formula for the nth term of a geometric sequence.

Understanding geometric sequences and their properties, particularly the common ratio, is crucial for solving various mathematical problems. The ability to predict terms in a sequence, calculate sums of series, and model real-world phenomena depends heavily on these concepts. The formula provides a systematic approach to tackle these problems, making it an essential tool in mathematics.

Mastering the formula and the underlying principles of geometric sequences not only aids in academic pursuits but also has practical applications in finance, science, and engineering. The concepts discussed in this article serve as a foundation for more advanced topics in mathematics, such as calculus, differential equations, and mathematical modeling. By grasping these fundamentals, students and professionals can better understand and solve complex problems in their respective fields. The significance of geometric sequences extends far beyond textbook exercises, offering valuable insights into the mathematical patterns that govern our world.