Geometric Sequence Recursive Formula The Fifth Term Is 781.25

Geometric sequences are a fundamental concept in mathematics, characterized by a constant ratio between consecutive terms. Understanding geometric sequences is crucial for various applications, from financial calculations to modeling natural phenomena. In this article, we will delve into a specific problem involving a geometric sequence, exploring how to determine its recursive formula.

Problem Statement

We are given that the fifth term of a geometric sequence is 781.25. A crucial piece of information is that each term of the sequence is [$\frac{1}{5}$] of the value of the following term. Our objective is to identify the recursive formula that accurately represents this situation. Understanding recursive formulas and their application to geometric sequences is key to solving this problem. A recursive formula defines a sequence by specifying the initial term(s) and a rule for calculating subsequent terms based on the preceding ones. This contrasts with an explicit formula, which directly calculates any term in the sequence without needing to know the previous terms. The recursive approach is particularly useful when the relationship between consecutive terms is readily apparent, as in this case where each term is a fraction of the next. By carefully analyzing the given information and applying the principles of geometric sequences, we can derive the recursive formula that precisely describes the sequence's behavior. This involves identifying the common ratio and the initial term(s) necessary to define the sequence recursively. The ability to translate the problem's conditions into a mathematical expression is a core skill in sequence analysis and is essential for determining the correct recursive representation. Let's explore the options provided and determine the correct recursive formula that represents the situation, ensuring a thorough understanding of the sequence's properties and its recursive definition.

Understanding Geometric Sequences

To effectively solve this problem, a solid grasp of geometric sequences is essential. A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a constant factor, known as the common ratio (r). This common ratio is the defining characteristic of a geometric sequence, dictating how the sequence progresses. The general form of a geometric sequence can be expressed as: a, ar, ar^2, ar^3, ... where 'a' represents the first term and 'r' is the common ratio. Each term in the sequence is derived by multiplying the preceding term by this constant ratio. This multiplicative relationship distinguishes geometric sequences from arithmetic sequences, where terms are generated by adding a constant difference. The common ratio can be any real number, including fractions and negative values, leading to diverse patterns within geometric sequences. For instance, if the common ratio is a fraction between 0 and 1, the terms of the sequence will decrease in magnitude, approaching zero. Conversely, if the common ratio is greater than 1, the terms will increase exponentially. A negative common ratio will result in an alternating sequence, where the signs of the terms alternate between positive and negative. Understanding the role of the common ratio is crucial for analyzing and predicting the behavior of geometric sequences, as it dictates the rate and direction of change within the sequence. In the context of this problem, the information that each term is a fraction of the following term directly relates to the common ratio, providing a key insight into the sequence's structure. By carefully considering the properties of geometric sequences and the significance of the common ratio, we can effectively approach the task of identifying the correct recursive formula.

Recursive Formulas for Geometric Sequences

A recursive formula provides a way to define a sequence by relating each term to the preceding term(s). In the case of a geometric sequence, the recursive formula typically takes the form: a_n = r * a_{n-1}, where a_n represents the nth term, a_{n-1} represents the (n-1)th term, and r is the common ratio. This formula states that any term in the sequence is equal to the common ratio multiplied by the previous term. To fully define a sequence recursively, it's not enough to just provide the recursive relationship; we also need to specify the initial term(s). For a geometric sequence, we usually need to specify the first term, a_1. This initial term acts as the starting point for the sequence, and the recursive formula then generates the subsequent terms. Combining the recursive relationship with the initial term(s) creates a complete definition of the sequence, allowing us to calculate any term in the sequence. Understanding the structure of recursive formulas is crucial for working with sequences, as it highlights the dependence of each term on its predecessors. This contrasts with explicit formulas, which provide a direct way to calculate any term without needing to know the previous terms. Recursive formulas are particularly useful when the relationship between consecutive terms is readily apparent, as in geometric sequences where each term is a constant multiple of the previous term. In the problem we are addressing, the given information about each term being a fraction of the next term directly points to the common ratio and the recursive relationship. By carefully applying the principles of recursive formulas and geometric sequences, we can determine the correct recursive formula that represents the given situation.

Analyzing the Given Information

We are given two key pieces of information: the fifth term of the geometric sequence is 781.25, and each term is [$\frac{1}{5}$] of the value of the following term. The first piece of information, a_5 = 781.25, provides a specific value within the sequence, which can be used to anchor our calculations and verify potential formulas. The second piece of information, that each term is [$\frac{1}{5}$] of the following term, is crucial for determining the common ratio (r) of the geometric sequence. This statement implies that to get from one term to the next, we multiply by [$\frac{1}{5}$]. Mathematically, this can be expressed as: a_n = [$\frac{1}{5}$] * a_n+1}. However, for a recursive formula, we need to express the nth term in terms of the (n-1)th term. To do this, we can rewrite the equation as a_{n+1 = 5 * a_n. This form tells us that each term is 5 times the previous term, which means the common ratio r is 5. By carefully analyzing the relationship between consecutive terms, we have identified the common ratio, a critical step in determining the recursive formula. This understanding allows us to translate the problem's description into a mathematical relationship, paving the way for constructing the correct recursive formula. Now, we need to determine the initial term (a_1) to complete the recursive definition of the sequence. This can be done by working backward from the fifth term using the common ratio, allowing us to fully characterize the sequence and its recursive representation. By combining the given information and applying the properties of geometric sequences, we can confidently identify the recursive formula that accurately describes the sequence.

Determining the Recursive Formula

Based on our analysis, we know that the common ratio (r) is 5. This means the recursive relationship can be written as: a_n = 5 * a_n-1}. Now, we need to find the first term (a_1) to complete the recursive formula. We know that the fifth term (a_5) is 781.25. To find a_1, we can work backward through the sequence, dividing by the common ratio (5) for each step. a_4 = a_5 / 5 = 781.25 / 5 = 156.25 a_3 = a_4 / 5 = 156.25 / 5 = 31.25 a_2 = a_3 / 5 = 31.25 / 5 = 6.25 a_1 = a_2 / 5 = 6.25 / 5 = 1.25 So, the first term (a_1) is 1.25. Now we have all the necessary components to write the recursive formula a_n = 5 * a_{n-1 a_1 = 1.25 This recursive formula defines the geometric sequence where each term is 5 times the previous term, and the first term is 1.25. By working backward from the given fifth term and utilizing the common ratio, we have successfully determined the initial term and constructed the complete recursive formula. This formula accurately represents the given geometric sequence, allowing us to calculate any term in the sequence by iteratively applying the recursive relationship. The process of finding the recursive formula highlights the importance of understanding the relationship between consecutive terms and the role of the initial term in defining a sequence. By combining these elements, we can effectively represent geometric sequences using recursive formulas.

Solution

The recursive formula that represents the situation is:

A. a_n = 5 * a_{n-1} ; a_1 = 1.25

This formula accurately captures the geometric sequence where each term is 5 times the previous term, and the first term is 1.25. We arrived at this solution by carefully analyzing the given information, identifying the common ratio, and working backward from the fifth term to determine the first term. This process demonstrates the power of recursive formulas in defining sequences and the importance of understanding the relationships between consecutive terms. By expressing the sequence recursively, we can easily calculate any term in the sequence by iteratively applying the formula. This approach provides a clear and concise representation of the sequence's behavior, highlighting its fundamental properties and allowing for further analysis and manipulation. The correct recursive formula effectively encapsulates the given information and provides a complete definition of the geometric sequence.