Finding The Recursive Formula For The Geometric Sequence 2, -10, 50, -250...

Delving into Geometric Sequences and Recursive Formulas

In the realm of mathematics, sequences play a crucial role in understanding patterns and relationships between numbers. Among these sequences, geometric sequences hold a special significance due to their consistent multiplicative nature. A geometric sequence is characterized by a constant ratio between consecutive terms, known as the common ratio. This fundamental property allows us to express any term in the sequence in terms of its preceding term, leading to the concept of a recursive formula.

Recursive formulas provide a powerful tool for defining sequences by establishing a relationship between a term and its predecessors. In essence, a recursive formula specifies the initial term(s) of the sequence and then provides a rule for generating subsequent terms based on the preceding ones. This approach is particularly useful for geometric sequences, where the common ratio dictates the multiplicative relationship between terms. To truly grasp the recursive formula for a geometric sequence, it's essential to understand the underlying principles of geometric sequences and how they relate to recursive definitions. Geometric sequences are sequences where each term is found by multiplying the previous term by a constant. This constant multiplier is known as the common ratio, often denoted by 'r'. For example, the sequence 2, 4, 8, 16... is a geometric sequence with a common ratio of 2, since each term is twice the previous term. Identifying the common ratio is the first step in deriving the recursive formula. A recursive formula defines a sequence by expressing each term as a function of the preceding term(s). This approach is particularly useful when there's a clear pattern connecting consecutive terms. The recursive formula typically consists of two parts the initial term(s) and the recursive equation. The initial term(s) establish the starting point of the sequence, while the recursive equation describes how to generate subsequent terms. For instance, the recursive formula for the sequence 2, 4, 8, 16... would specify the first term (a₁ = 2) and the recursive equation (an = 2 * an-1), indicating that each term is twice the previous term. Understanding these core concepts lays the foundation for unraveling the recursive formula for the given geometric sequence.

Decoding the Given Geometric Sequence: 2, -10, 50, -250...

Now, let's turn our attention to the specific geometric sequence presented: 2, -10, 50, -250, .... Our mission is to decipher the underlying pattern and express it in the form of a recursive formula. To achieve this, we must first identify the common ratio that governs the sequence. By examining the consecutive terms, we can observe the multiplicative relationship between them. Dividing the second term (-10) by the first term (2) yields -5. Similarly, dividing the third term (50) by the second term (-10) also gives us -5. This consistent ratio indicates that the sequence is indeed geometric, with a common ratio of -5. The common ratio is the key to unlocking the recursive formula. Once we've determined the common ratio, we can express each term as the product of the previous term and this ratio. In this case, each term is -5 times the previous term. This observation forms the basis of the recursive equation. The first term is also a crucial component of the recursive formula. It serves as the starting point for generating the sequence. In our sequence, the first term (a₁) is 2. This value will be explicitly stated in the recursive formula. With the common ratio and the first term in hand, we have all the necessary ingredients to construct the recursive formula. The formula will consist of two parts the initial condition, specifying the first term, and the recursive equation, defining the relationship between consecutive terms. By carefully combining these elements, we can accurately represent the given geometric sequence in a recursive manner.

Constructing the Recursive Formula: A Step-by-Step Approach

With the common ratio and the first term identified, we are now ready to assemble the recursive formula. As mentioned earlier, the recursive formula consists of two essential parts: the initial condition and the recursive equation. The initial condition simply states the value of the first term, which in our case is 2. This can be written as: a₁ = 2. This sets the stage for the sequence, providing the starting point for generating subsequent terms. The recursive equation captures the relationship between a term and its predecessor. In our geometric sequence, each term is obtained by multiplying the previous term by the common ratio, which is -5. Therefore, the recursive equation can be expressed as: an = an-1 * (-5). This equation elegantly encapsulates the multiplicative nature of the sequence, specifying how to obtain any term (an) based on its preceding term (an-1). Combining the initial condition and the recursive equation, we arrive at the complete recursive formula for the given geometric sequence: 1. a₁ = 2 2. an = an-1 * (-5). This formula provides a concise and accurate representation of the sequence, allowing us to generate any term in the sequence by iteratively applying the recursive equation. To illustrate how the recursive formula works, let's generate the first few terms of the sequence. We start with the initial term, a₁ = 2. To find the second term (a₂), we substitute n = 2 into the recursive equation: a₂ = a₁ * (-5) = 2 * (-5) = -10. Similarly, to find the third term (a₃), we substitute n = 3: a₃ = a₂ * (-5) = -10 * (-5) = 50. Continuing this process, we can generate the entire sequence, confirming that the recursive formula accurately represents the given geometric sequence.

Evaluating the Answer Choices: Identifying the Correct Recursive Formula

Now that we have derived the recursive formula for the geometric sequence, let's evaluate the given answer choices to identify the correct one. The answer choices are presented in the following format:

A. $\left\{\begin{array}{l}a_1=2 \\ a_n=a_{n-1}\end{array} \bullet(-8)\right.$

B. $\left\{\begin{array}{l}a_1=2 \\ a_n=a_{n-1}\end{array} \bullet(-5)\right.$

Comparing these options with the recursive formula we derived, we can see that option B closely matches our result. Option B states: 1. a₁ = 2 2. an = an-1 * (-5). This is precisely the recursive formula we constructed, with the correct initial term and the recursive equation reflecting the common ratio of -5. Option A, on the other hand, has an incorrect recursive equation. It suggests that each term is obtained by multiplying the previous term by -8, which is not the common ratio of our sequence. Therefore, option A is incorrect. By carefully comparing the answer choices with our derived recursive formula, we can confidently conclude that option B is the correct answer. This demonstrates the importance of understanding the underlying principles of geometric sequences and recursive formulas, as it allows us to not only derive the formula but also verify its correctness.

Conclusion: The Power of Recursive Formulas in Describing Geometric Sequences

In conclusion, the recursive formula for the geometric sequence 2, -10, 50, -250, ... is given by:

1. a₁ = 2
2. an = an-1 * (-5)

This formula elegantly captures the essence of the geometric sequence, defining each term in relation to its predecessor. The initial condition (a₁ = 2) establishes the starting point, while the recursive equation (an = an-1 * (-5)) embodies the multiplicative relationship governed by the common ratio of -5.

Understanding recursive formulas is crucial for working with geometric sequences and other mathematical patterns. They provide a concise and powerful way to define sequences, allowing us to generate terms and analyze their behavior. By mastering the concepts of common ratios, initial conditions, and recursive equations, we can effectively unravel the structure of geometric sequences and express them in a recursive form. This understanding not only aids in solving specific problems but also enhances our overall mathematical reasoning and problem-solving skills.

This exploration of the recursive formula for the given geometric sequence highlights the beauty and power of mathematical tools in describing and understanding patterns in the world around us. From simple sequences to complex mathematical models, recursive relationships play a fundamental role in capturing the essence of change and progression. As we continue our mathematical journey, the insights gained from studying recursive formulas will undoubtedly prove invaluable in tackling a wide range of problems and challenges.

Therefore, the correct answer is B.

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"title": "Finding the Recursive Formula for the Geometric Sequence 2, -10, 50, -250..."