Geometric Sequences Common Ratio In Explicit Formula

Hey guys! Let's dive into the fascinating world of sequences, specifically geometric sequences. These sequences are super cool because they follow a predictable pattern, where each term is found by multiplying the previous term by a constant value. This constant value is what we call the common ratio, and it's the key to understanding and working with geometric sequences.

In this article, we're going to tackle a specific problem that involves finding the common ratio of a geometric sequence. We'll break down the problem step by step, making sure you understand the logic behind each step. By the end, you'll be a pro at identifying common ratios and using them to solve sequence-related problems. So, let's get started!

Understanding Geometric Sequences

Before we jump into the problem, let's make sure we're all on the same page about what a geometric sequence actually is. A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed, non-zero number. This fixed number is the common ratio. Think of it like this: you start with a number, and then you consistently multiply by the same value to get the next number in the sequence. This consistent multiplication is what defines a geometric sequence.

For example, the sequence 2, 4, 8, 16, 32... is a geometric sequence. Can you spot the common ratio? It's 2! Each term is twice the previous term (2 * 2 = 4, 4 * 2 = 8, and so on). Understanding this basic concept is crucial for solving problems related to geometric sequences.

Key characteristics of geometric sequences include:

• Constant Ratio: The ratio between consecutive terms remains the same throughout the sequence. This is the defining feature of a geometric sequence.
• Exponential Growth or Decay: Geometric sequences exhibit either exponential growth (if the common ratio is greater than 1) or exponential decay (if the common ratio is between 0 and 1).
• Explicit Formula: We can express any term in a geometric sequence using an explicit formula, which directly relates the term to its position in the sequence. This formula is incredibly useful for finding specific terms without having to calculate all the preceding terms.

Explicit Formulas: The Key to Unlocking Geometric Sequences

Explicit formulas are like secret codes that allow us to directly calculate any term in a sequence without having to know the previous terms. For a geometric sequence, the explicit formula takes a specific form that involves the first term, the common ratio, and the term's position in the sequence. Let's break down the formula and see how it works.

The general explicit formula for a geometric sequence is:

a_n = a_1 * r^(n-1)

Where:

• a_n is the nth term in the sequence (the term we want to find).
• a_1 is the first term in 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 might look a bit intimidating at first, but it's actually quite straightforward. It tells us that to find any term in a geometric sequence, we simply multiply the first term by the common ratio raised to the power of (n-1), where n is the term number.

Let's take our previous example, the sequence 2, 4, 8, 16, 32..., and see how the explicit formula works. We know that the first term (a_1) is 2 and the common ratio (r) is 2. Let's say we want to find the 5th term (a_5). Using the formula:

a_5 = 2 * 2^(5-1) = 2 * 2^4 = 2 * 16 = 32

As you can see, the formula correctly gives us the 5th term, which is 32. This demonstrates the power of the explicit formula in determining any term in a geometric sequence without having to list out all the preceding terms.

Diving into the Problem: Finding the Common Ratio

Now that we have a solid understanding of geometric sequences and explicit formulas, let's tackle the problem at hand. The problem states that a sequence begins with 12 and progresses geometrically. This means that each term is obtained by multiplying the previous term by a constant common ratio. We're also told that each number is the previous number divided by 2. This is a crucial piece of information that will help us find the common ratio.

Remember, dividing by a number is the same as multiplying by its reciprocal. So, dividing by 2 is the same as multiplying by 1/2. This gives us a strong clue about the common ratio.

The question asks us which value can be used as the common ratio in an explicit formula that represents the sequence. We're given two options:

A. 1/2 B. 2

Let's analyze these options based on what we know about the sequence.

Solving the Puzzle: Identifying the Correct Common Ratio

We know that the sequence starts with 12, and each subsequent term is obtained by dividing the previous term by 2. This means the sequence looks something like this: 12, 6, 3, 1.5...

To find the common ratio, we can simply divide any term by its preceding term. For example, let's divide the second term (6) by the first term (12):

Common Ratio = 6 / 12 = 1/2

Alternatively, we could divide the third term (3) by the second term (6):

Common Ratio = 3 / 6 = 1/2

In both cases, we get the same common ratio: 1/2. This confirms our earlier suspicion that the common ratio should be 1/2, since dividing by 2 is equivalent to multiplying by 1/2.

Now, let's consider the options provided in the problem:

A. 1/2 B. 2

Based on our calculations, we can confidently say that the correct answer is A. 1/2. The value 1/2 can be used as the common ratio in an explicit formula that represents the sequence.

Option B, 2, is incorrect. If the common ratio were 2, the sequence would be multiplying by 2, not dividing. This would result in a sequence that grows larger with each term (12, 24, 48...), which is not what the problem describes.

Crafting the Explicit Formula: Putting It All Together

Now that we've found the common ratio, let's take it a step further and write the explicit formula for this geometric sequence. This will solidify our understanding of how the common ratio fits into the formula and how we can use it to find any term in the sequence.

Remember, the general explicit formula for a geometric sequence is:

a_n = a_1 * r^(n-1)

We know the first term (a_1) is 12, and we've determined that the common ratio (r) is 1/2. So, we can plug these values into the formula:

a_n = 12 * (1/2)^(n-1)

This is the explicit formula that represents the sequence 12, 6, 3, 1.5... Let's test it out to make sure it works. Let's say we want to find the 4th term (a_4). Using the formula:

a_4 = 12 * (1/2)^(4-1) = 12 * (1/2)^3 = 12 * (1/8) = 1.5

The formula correctly gives us the 4th term, which is 1.5. This confirms that our explicit formula is accurate and can be used to find any term in the sequence.

The explicit formula allows us to directly calculate any term without having to know the preceding terms, making it a powerful tool for working with geometric sequences. This formula encapsulates the essence of the geometric sequence: the starting value (first term) and the consistent multiplicative factor (common ratio).

Key Takeaways and Conclusion

We've covered a lot in this article, guys! Let's recap the key takeaways:

• Geometric sequences are sequences where each term is obtained by multiplying the previous term by a constant common ratio.
• The common ratio is the key to understanding and working with geometric sequences.
• The explicit formula for a geometric sequence is a_n = a_1 * r^(n-1), where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the term's position.
• Dividing by a number is the same as multiplying by its reciprocal, which can help you identify the common ratio when dealing with division in a sequence.

By understanding these concepts, you'll be well-equipped to tackle a wide range of problems involving geometric sequences. Remember to always look for the pattern and identify the common ratio, as this is the foundation for solving these types of problems.

In this specific problem, we successfully identified the common ratio as 1/2 by recognizing that dividing by 2 is the same as multiplying by 1/2. We then used this common ratio to construct the explicit formula for the sequence. This demonstrates the power of understanding the fundamental principles of geometric sequences and how they can be applied to solve problems.

So, keep practicing, keep exploring, and you'll become a geometric sequence master in no time! Remember, math is all about understanding the underlying concepts and applying them in creative ways. And you guys are definitely capable of doing just that!