Finding The Explicit Formula For A Geometric Sequence A Step-by-Step Guide

by ADMIN 75 views
Iklan Headers

Hey guys! Today, we're diving deep into the fascinating world of geometric sequences and their explicit formulas. If you've ever been stumped by a sequence of numbers and wondered how to find any term in that sequence without manually calculating each one, then you're in the right place. We're going to break down what explicit formulas are, how they work, and how to identify the correct one for a given geometric sequence. So, buckle up and let's get started!

Understanding Geometric Sequences

Before we jump into explicit formulas, let's quickly recap what geometric sequences are. In geometric sequences, each term is obtained by multiplying the previous term by a constant value, which we call the common ratio. Think of it like a chain reaction where each number is linked to the one before it by the same multiplication factor. For instance, the sequence 2, 6, 18, 54,... is a geometric sequence because each term is three times the previous term (the common ratio is 3). Understanding this fundamental concept is crucial for grasping how explicit formulas work.

The first step in working with a geometric sequence is always identifying the common ratio. This is the constant factor by which each term is multiplied to obtain the next term. To find the common ratio, simply divide any term by its preceding term. Let's illustrate this with an example. Suppose we have the sequence 4, 8, 16, 32, ... To find the common ratio, we can divide 8 by 4, 16 by 8, or 32 by 16, and in each case, we get 2. This tells us that the common ratio for this sequence is 2. Once you've identified the common ratio, you're one step closer to writing the explicit formula.

Another important aspect of geometric sequences is the first term. The first term is simply the initial value of the sequence, and it plays a critical role in defining the entire sequence. In our example sequence 4, 8, 16, 32, ..., the first term is 4. The first term, along with the common ratio, uniquely defines a geometric sequence, and these two values are the key ingredients in constructing the explicit formula. Understanding how the first term and common ratio interact is essential for predicting future terms and analyzing the overall behavior of the sequence.

What are Explicit Formulas?

So, what exactly is an explicit formula? An explicit formula is a mathematical equation that allows you to directly calculate any term in a sequence (or a geometric sequence) without having to know the previous terms. It's like a magic key that unlocks the value of any term you desire, simply by plugging in its position in the sequence. Explicit formulas are incredibly powerful tools because they provide a concise and efficient way to describe and work with sequences. Imagine you wanted to find the 100th term of a sequence – with an explicit formula, you can calculate it directly, without having to compute the first 99 terms.

The general form of an explicit formula for a geometric sequence is given by:

a_n = a_1 * r^(n-1)

Where:

  • a_n is the nth term of the sequence (the term we want to find).
  • a_1 is the first term of the sequence.
  • r is the common ratio.
  • n is the position of the term in the sequence (e.g., 1 for the first term, 2 for the second term, and so on).

This formula encapsulates the essence of a geometric sequence: each term is the product of the first term and the common ratio raised to a power that corresponds to its position in the sequence. The (n-1) exponent is crucial because it accounts for the fact that the first term already exists, and we're multiplying by the common ratio to get the subsequent terms. Let's break down this formula with an example. Consider the sequence 3, 6, 12, 24,... The first term (a_1) is 3, and the common ratio (r) is 2. To find the 5th term (a_5), we plug these values into the explicit formula:

a_5 = 3 * 2^(5-1) = 3 * 2^4 = 3 * 16 = 48

So, the 5th term of this sequence is 48. This demonstrates the power and efficiency of explicit formulas in determining any term in a geometric sequence.

Analyzing the Given Geometric Sequence

Now, let's apply our knowledge to the specific geometric sequence in question: 0.5, -0.1, 0.02, -0.004, 0.0008, ... Our goal is to find the explicit formula that accurately represents this sequence. To do this, we need to identify the first term (a_1) and the common ratio (r).

First, let's identify the first term. Looking at the sequence, the first term (a_1) is clearly 0.5. This is the starting point of our sequence, and it's the foundation upon which all subsequent terms are built. The first term is a crucial component of the explicit formula, as it serves as the initial value that is repeatedly multiplied by the common ratio.

Next, we need to determine the common ratio. To find the common ratio, we can divide any term by its preceding term. Let's divide the second term (-0.1) by the first term (0.5):

r = -0.1 / 0.5 = -0.2

We can verify this by dividing the third term (0.02) by the second term (-0.1), which also gives us -0.2. This confirms that our common ratio (r) is indeed -0.2. The common ratio is the engine that drives the sequence, determining how each term changes relative to the previous one. In this case, the negative common ratio indicates that the terms will alternate in sign, while the magnitude of 0.2 indicates that the terms will decrease in absolute value.

Now that we've identified the first term (a_1 = 0.5) and the common ratio (r = -0.2), we have all the pieces we need to construct the explicit formula for this geometric sequence. We can now substitute these values into the general form of the explicit formula and compare our result with the given options.

Evaluating the Answer Choices

Alright, we've done the groundwork – we've identified the first term (0.5) and the common ratio (-0.2) for our geometric sequence. Now, it's time to put on our detective hats and evaluate the given answer choices to see which one matches our findings. Remember, the explicit formula should have the form a_n = a_1 * r^(n-1), where a_1 is the first term, r is the common ratio, and n is the term number.

Let's take a look at the answer choices:

A. a_n = 0.5(0.2)^n

B. a_n = -0.5(-0.2)^(n-1)

C. a_n = 0.5(-0.2)^(n-1)

To determine the correct answer, we'll compare each option to the general form of the explicit formula and see if it matches our calculated values for a_1 and r. We'll also pay close attention to the exponent and the signs.

Let's start with option A: a_n = 0.5(0.2)^n. This formula has the correct first term (0.5), but the common ratio is positive (0.2) instead of negative (-0.2), and the exponent is n instead of n-1. So, this option is not the correct one.

Now, let's consider option B: a_n = -0.5(-0.2)^(n-1). This formula has a negative first term (-0.5), which doesn't match our identified first term of 0.5. While the common ratio (-0.2) and the exponent (n-1) are correct, the incorrect first term disqualifies this option.

Finally, let's examine option C: a_n = 0.5(-0.2)^(n-1). This formula has the correct first term (0.5), the correct common ratio (-0.2), and the correct exponent (n-1). This option perfectly matches the general form of the explicit formula and our calculated values. Therefore, option C is the correct answer.

By carefully evaluating each answer choice and comparing it to our calculated values, we were able to confidently identify the correct explicit formula for the given geometric sequence. This process highlights the importance of understanding the components of the explicit formula and how they relate to the characteristics of the sequence.

Why Option C is the Correct Explicit Formula

So, we've pinpointed option C (a_n = 0.5(-0.2)^(n-1)) as the correct explicit formula, but let's solidify our understanding by delving into why it's the right choice. We'll break down each component of the formula and see how it accurately represents the geometric sequence 0.5, -0.1, 0.02, -0.004, 0.0008, ...

The formula a_n = 0.5(-0.2)^(n-1) has three key components:

  1. The first term (0.5): This is the initial value of the sequence, and it's correctly represented in the formula as the coefficient multiplying the exponential term. This ensures that when n = 1, the formula yields the first term of the sequence, which is indeed 0.5.
  2. The common ratio (-0.2): This is the constant factor by which each term is multiplied to obtain the next term. The formula accurately captures this with the base of the exponential term being -0.2. The negative sign indicates that the terms will alternate in sign, and the magnitude of 0.2 determines the rate at which the terms decrease in absolute value.
  3. The exponent (n-1): This exponent ensures that the common ratio is raised to the appropriate power for each term. When n = 1, the exponent is 0, so the common ratio is raised to the power of 0, resulting in 1. This means that the first term is simply 0.5 (as it should be). For subsequent terms, the exponent increases by 1 for each position, effectively multiplying the previous term by the common ratio.

To further illustrate why option C is correct, let's calculate the first few terms using the formula:

  • For n = 1: a_1 = 0.5(-0.2)^(1-1) = 0.5(-0.2)^0 = 0.5 * 1 = 0.5
  • For n = 2: a_2 = 0.5(-0.2)^(2-1) = 0.5(-0.2)^1 = 0.5 * -0.2 = -0.1
  • For n = 3: a_3 = 0.5(-0.2)^(3-1) = 0.5(-0.2)^2 = 0.5 * 0.04 = 0.02

As you can see, the formula correctly generates the first three terms of the sequence. This pattern continues for all terms, confirming that option C is indeed the correct explicit formula.

By understanding how each component of the formula contributes to the overall sequence, we can appreciate the elegance and efficiency of explicit formulas in describing geometric sequences.

Key Takeaways for Mastering Explicit Formulas

We've journeyed through the world of explicit formulas for geometric sequences, and hopefully, you're feeling more confident in your ability to tackle these problems. Before we wrap up, let's highlight some key takeaways that will help you master explicit formulas:

  1. Understand Geometric Sequences: The foundation of working with explicit formulas is a solid understanding of geometric sequences. Remember that each term is obtained by multiplying the previous term by a constant common ratio. Identifying the common ratio is the first crucial step in finding the explicit formula.
  2. Master the General Form: Memorize the general form of the explicit formula for a geometric sequence: a_n = a_1 * r^(n-1). Understanding this formula and its components will make it much easier to construct and identify explicit formulas.
  3. Identify the First Term and Common Ratio: The first term (a_1) and the common ratio (r) are the building blocks of the explicit formula. Practice identifying these values from a given sequence. Remember to find the common ratio by dividing any term by its preceding term.
  4. Pay Attention to Signs: The sign of the common ratio is crucial. A positive common ratio indicates that the terms will have the same sign, while a negative common ratio indicates that the terms will alternate in sign. Be mindful of this when evaluating answer choices.
  5. Evaluate Answer Choices Carefully: When presented with multiple answer choices, systematically evaluate each one by comparing it to the general form of the explicit formula and your calculated values for the first term and common ratio. Substitute values of n to check if the formula generates the correct terms.
  6. Practice, Practice, Practice: The key to mastering any mathematical concept is practice. Work through various examples of geometric sequences and explicit formulas to solidify your understanding. The more you practice, the more comfortable and confident you'll become.

By keeping these takeaways in mind and consistently practicing, you'll be well on your way to becoming a pro at working with explicit formulas for geometric sequences. So, go forth and conquer those sequences!

Conclusion

Alright guys, we've reached the end of our explicit formula adventure! We started by understanding the fundamentals of geometric sequences and what makes them tick. Then, we dove into the world of explicit formulas, learning how they act as a secret code to unlock any term in a sequence. We dissected the given sequence, identified its key ingredients (the first term and common ratio), and meticulously evaluated the answer choices. And finally, we crowned the correct explicit formula, option C, as the winner! Remember, practice makes perfect, so keep those sequences coming and those formulas flowing. You've got this!