Exponential Function For Geometric Sequence F(n) = 1,250(11)^(n-1)

by Alex Johnson 67 views

Understanding geometric sequences and their exponential function representations is a fundamental concept in mathematics. When presented with an explicit formula for a geometric sequence, such as f(n) = 1,250(11)^(n-1), the task of determining the exponential function might seem daunting at first. However, by breaking down the components of the formula and understanding the underlying principles of geometric sequences, we can easily unravel the exponential function. This article will guide you through the process, ensuring you grasp the core concepts and can apply them to similar problems.

Understanding Geometric Sequences and Explicit Formulas

Before diving into the specifics of our example, let's establish a solid foundation by defining what a geometric sequence is and how its explicit formula works. A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a constant value, known as the common ratio. This contrasts with arithmetic sequences, where a constant difference is added between terms. The beauty of geometric sequences lies in their consistent multiplicative pattern, which allows us to predict future terms with ease.

The explicit formula for a geometric sequence provides a direct way to calculate any term in the sequence without needing to know the preceding terms. It generally takes the form:

  • f(n) = a * r^(n-1)

Where:

  • f(n) is the nth term of the sequence
  • a is the first term of the sequence
  • r is the common ratio
  • n is the term number

This formula encapsulates the essence of a geometric sequence, highlighting the role of the first term and the common ratio in shaping the sequence's progression. Understanding this formula is crucial for both analyzing and constructing geometric sequences.

In our specific case, we are given the explicit formula f(n) = 1,250(11)^(n-1). By comparing this to the general form, we can immediately identify that the first term, a, is 1,250, and the common ratio, r, is 11. This tells us that the sequence starts at 1,250, and each subsequent term is 11 times larger than the previous one. This information is vital for constructing the exponential function that represents this sequence.

Deconstructing the Given Formula: f(n) = 1,250(11)^(n-1)

Now, let's dissect the given explicit formula, f(n) = 1,250(11)^(n-1), to truly understand its components and how they contribute to the sequence. As we identified earlier, 1,250 represents the first term (a) of the sequence. This is the starting point, the anchor from which the sequence grows. The number 11 is the common ratio (r), the engine driving the sequence's exponential growth. Each term is 11 times the previous term, leading to rapid increases as n increases. The term (n-1) in the exponent is crucial because it ensures that when n is 1 (the first term), the exponent becomes 0, and 11^0 equals 1, leaving us with just the first term, 1,250. This is a clever way to align the formula with the sequence's starting point.

To further illustrate, let's calculate the first few terms of the sequence using the formula:

  • f(1) = 1,250(11)^(1-1) = 1,250(11)^0 = 1,250 * 1 = 1,250
  • f(2) = 1,250(11)^(2-1) = 1,250(11)^1 = 1,250 * 11 = 13,750
  • f(3) = 1,250(11)^(3-1) = 1,250(11)^2 = 1,250 * 121 = 151,250

As you can see, the sequence grows dramatically with each term. This exponential growth is a hallmark of geometric sequences, and it's precisely what makes them so useful in modeling various real-world phenomena, from compound interest to population growth.

The exponent (n-1) plays a critical role in defining the sequence's behavior. It dictates how quickly the terms increase and maintains the correct relationship between the term number and its value. Without this exponent, the sequence would not be geometric; it would simply be a constant value multiplied by n. By understanding the significance of each component in the formula, we gain a deeper appreciation for the elegance and power of geometric sequences.

Transforming the Explicit Formula into an Exponential Function

The heart of the matter lies in transforming the explicit formula into its corresponding exponential function. The explicit formula f(n) = 1,250(11)^(n-1) is a discrete function, meaning it's defined only for integer values of n (1, 2, 3, and so on). An exponential function, on the other hand, is a continuous function, defined for all real numbers. To make this transformation, we need to shift our perspective slightly.

Recall that an exponential function generally takes the form:

  • f(x) = a * b^x

Where:

  • f(x) is the value of the function at x
  • a is the initial value (the value when x is 0)
  • b is the base, representing the growth or decay factor
  • x is the independent variable

Notice the similarity between this form and the explicit formula. The key difference is that in the exponential function, the exponent is simply x, while in the explicit formula, it's (n-1). To bridge this gap, we need to manipulate the explicit formula algebraically.

Starting with f(n) = 1,250(11)^(n-1), we can use the properties of exponents to rewrite the expression:

  • f(n) = 1,250(11)^n * (11)^(-1)

Remember that x^(a-b) = x^a * x^(-b). Now, we can simplify further:

  • f(n) = 1,250(11)^n * (1/11)
  • f(n) = (1,250/11)(11)^n

Now, we have an expression that closely resembles the standard form of an exponential function. We can see that the base, b, is 11, which is the same as the common ratio in the explicit formula. The initial value, a, is 1,250/11, which is approximately 113.64. This value represents the function's value when n (or x) is 0. Therefore, the exponential function that represents the given geometric sequence is:

  • f(x) = (1,250/11)(11)^x

This function captures the essence of the geometric sequence, allowing us to calculate the value at any real number x, not just integers. It provides a continuous representation of the discrete sequence, which can be useful in various applications.

Verifying the Exponential Function

To ensure our exponential function is correct, we can verify it by plugging in integer values for x and comparing the results to the terms of the original geometric sequence. Let's test a few values:

  • f(1) = (1,250/11)(11)^1 = 1,250 (matches the first term)
  • f(2) = (1,250/11)(11)^2 = 1,250 * 11 = 13,750 (matches the second term)
  • f(3) = (1,250/11)(11)^3 = 1,250 * 11^2 = 151,250 (matches the third term)

The exponential function accurately reproduces the terms of the geometric sequence for integer values of x. This confirms that our transformation was successful and that we have indeed found the correct exponential function.

Moreover, the exponential function provides us with insights beyond the discrete terms of the sequence. We can now analyze the function's behavior for non-integer values of x, gaining a more complete understanding of the growth pattern. For instance, we can calculate the value of the function at x = 1.5, which would represent a point in between the first and second terms of the sequence. This is a powerful capability that is not available with the explicit formula alone.

By verifying our result, we not only gain confidence in our solution but also reinforce our understanding of the relationship between explicit formulas and exponential functions. This process highlights the importance of checking our work and ensuring that our mathematical models accurately represent the situations they are intended to describe.

Applications and Implications

The ability to transform explicit formulas into exponential functions has significant implications in various fields. Geometric sequences and exponential functions are fundamental tools for modeling phenomena that exhibit exponential growth or decay. Understanding how to manipulate these mathematical constructs allows us to analyze and predict real-world events with greater accuracy.

In finance, geometric sequences are used to calculate compound interest. The explicit formula can determine the future value of an investment after a certain number of periods, while the exponential function provides a continuous view of the investment's growth over time. This allows investors to make informed decisions about their financial strategies.

In biology, exponential functions model population growth. The common ratio represents the rate at which a population increases, and the exponential function can predict the population size at any given time. This is crucial for understanding ecological dynamics and managing resources effectively.

In physics, radioactive decay is modeled using exponential functions. The decay constant determines the rate at which a radioactive substance decays, and the exponential function predicts the amount of substance remaining after a certain period. This is essential for applications in nuclear medicine and nuclear energy.

The transformation from an explicit formula to an exponential function provides a more versatile tool for analysis and prediction. While the explicit formula gives us specific terms of a sequence, the exponential function gives us a continuous representation, allowing us to interpolate and extrapolate values beyond the discrete terms. This is particularly valuable when dealing with real-world phenomena that change continuously over time.

Furthermore, the exponential function provides insights into the underlying growth or decay rate. The base of the exponential function, b, directly reflects this rate. If b is greater than 1, the function represents exponential growth; if b is between 0 and 1, it represents exponential decay. This information is crucial for understanding the long-term behavior of the modeled phenomenon.

In conclusion, the ability to convert between explicit formulas and exponential functions is a valuable skill with wide-ranging applications. It allows us to model and analyze exponential phenomena in various fields, providing insights and predictions that are essential for informed decision-making. By mastering this transformation, we gain a deeper understanding of the power and versatility of mathematical models.

Conclusion

In summary, determining the exponential function from the explicit formula of a geometric sequence involves understanding the components of the formula, manipulating it algebraically, and verifying the result. Starting with f(n) = 1,250(11)^(n-1), we identified the first term and common ratio, rewrote the formula using exponent properties, and arrived at the exponential function f(x) = (1,250/11)(11)^x. This process not only provides the answer but also reinforces the fundamental concepts of geometric sequences and exponential functions.

By following these steps, you can confidently tackle similar problems and gain a deeper appreciation for the beauty and power of mathematical relationships. Remember to always check your work and consider the implications of your results in real-world contexts. The connection between discrete sequences and continuous functions is a cornerstone of mathematical modeling, and mastering this connection will serve you well in various disciplines.

For further exploration of exponential functions and geometric sequences, consider visiting Khan Academy's section on exponential growth and decay.