Explicit Formula For Arithmetic Sequence -7.5, -9, -10.5, -12

Introduction to Arithmetic Sequences

In the realm of mathematics, arithmetic sequences hold a significant position as they represent a fundamental pattern of numbers. These sequences are characterized by a constant difference between consecutive terms, making them predictable and easily analyzable. Understanding arithmetic sequences is crucial for various applications, ranging from financial modeling to computer science. In this comprehensive guide, we will delve into the intricacies of arithmetic sequences, focusing on how to determine the explicit formula that defines them. This will be achieved by carefully examining the given sequence: -7.5, -9, -10.5, -12, ... Our objective is to derive the explicit formula that accurately represents this sequence, enabling us to find any term within it without having to list out all the preceding terms. This is particularly useful when dealing with sequences that extend to a large number of terms, where manually calculating each term would be impractical. The explicit formula provides a concise and efficient way to determine any term in the sequence, making it an invaluable tool for mathematicians and anyone working with numerical patterns. By the end of this exploration, you will have a solid grasp of how to identify arithmetic sequences, calculate the common difference, and construct the explicit formula. This knowledge will not only help you in solving similar problems but also enhance your overall understanding of mathematical sequences and their applications.

Identifying the Common Difference

To effectively decipher an arithmetic sequence, identifying the common difference is a pivotal first step. This constant difference, denoted as 'd', is the cornerstone of arithmetic sequences, dictating the consistent pattern between successive terms. To calculate the common difference, we subtract any term from its immediate successor. This simple yet crucial operation reveals the constant increment or decrement that defines the sequence. In our specific case, the sequence is given as -7.5, -9, -10.5, -12, ... To find the common difference, we can subtract the first term from the second term, the second from the third, and so on. Let's perform these calculations: -9 - (-7.5) = -1.5 -10.5 - (-9) = -1.5 -12 - (-10.5) = -1.5 As we can see, the difference between consecutive terms consistently equals -1.5. This confirms that the sequence is indeed arithmetic, and the common difference, d, is -1.5. Understanding the common difference is paramount because it not only confirms the arithmetic nature of the sequence but also serves as a key component in formulating the explicit formula. This formula, as we will see, leverages the common difference and the first term to define any term in the sequence. Recognizing and accurately calculating the common difference is therefore an essential skill in working with arithmetic sequences. This foundation will allow us to confidently move forward and construct the explicit formula that describes the sequence.

Constructing the Explicit Formula

Having established the common difference, the next crucial step is constructing the explicit formula. This formula serves as a mathematical blueprint, enabling us to directly calculate any term in the sequence without needing to know the preceding terms. The general form of an explicit formula for an arithmetic sequence is expressed as: a_n = a_1 + (n - 1)d where: a_n represents the nth term in the sequence, a_1 denotes the first term of the sequence, n is the position of the term we want to find, and d is the common difference. This formula is derived from the fundamental principle of arithmetic sequences: each term is obtained by adding the common difference to the previous term. The (n - 1) factor accounts for the number of times the common difference is added to the first term to reach the nth term. In our specific sequence, -7.5, -9, -10.5, -12, ..., we have already identified that the first term, a_1, is -7.5 and the common difference, d, is -1.5. Substituting these values into the general formula, we get: a_n = -7.5 + (n - 1)(-1.5) This is the explicit formula for the given arithmetic sequence. It allows us to find any term in the sequence by simply plugging in the value of n. For instance, to find the 10th term, we would substitute n = 10 into the formula. This ability to directly calculate terms makes the explicit formula a powerful tool for analyzing and working with arithmetic sequences.

Analyzing the Given Options

Now, let's analyze the given options and meticulously compare them with the explicit formula we derived. This step is crucial to ensure we select the correct representation of the sequence. The explicit formula we found is: a_n = -7.5 + (n - 1)(-1.5) We will now examine each option provided and determine which one matches our derived formula. This involves carefully comparing the structure of each option, paying close attention to the initial term, the common difference, and the way these components are combined within the formula. By systematically evaluating each option, we can confidently identify the one that accurately represents the arithmetic sequence -7.5, -9, -10.5, -12, ... This process not only helps us in selecting the correct answer but also reinforces our understanding of how explicit formulas are constructed and interpreted. Let's proceed to examine the given options:

Option 1: $a_n=-7.5+(-1.5)(n-1)$

This option, a_n = -7.5 + (-1.5)(n - 1), appears to be a direct match to the explicit formula we derived. It correctly identifies the first term as -7.5 and the common difference as -1.5, incorporating them into the formula in the standard format for arithmetic sequences. The (n - 1) term accurately represents the number of times the common difference is added to the first term to reach the nth term. Therefore, this option seems to be a strong candidate for the correct explicit formula. To further validate this, we can substitute a few values of n (e.g., n = 1, 2, 3) into the formula and check if the resulting terms match the given sequence. This will provide additional assurance that this option accurately represents the sequence. Given its initial alignment with our derived formula, this option holds a high probability of being the correct answer.

Option 2: $a_n=-7.5+a(-1.5-1)$

This option, a_n = -7.5 + a(-1.5 - 1), presents a significant deviation from the standard form of an explicit formula for arithmetic sequences. The presence of 'a' within the formula, without a clear indication of what it represents, is a major point of concern. Additionally, the expression (-1.5 - 1) simplifies to -2.5, which does not align with the common difference we calculated earlier (-1.5). The absence of the (n - 1) term, which is crucial for representing the position of the term in the sequence, further undermines the validity of this option. These factors collectively suggest that this option is unlikely to be the correct explicit formula for the given arithmetic sequence. The ambiguous 'a' term and the incorrect representation of the common difference make this option inconsistent with the fundamental principles of arithmetic sequences and their explicit formulas. Therefore, we can confidently rule out this option as a potential answer.

Option 3: $a_n=-1.5+(-7.5)(n-1)$

This option, a_n = -1.5 + (-7.5)(n - 1), presents a different arrangement of the components but still needs careful evaluation. While it includes the (n - 1) term, which is essential for an explicit formula, it incorrectly identifies the first term and the common difference. In this option, -1.5 is presented as the first term, and -7.5 is multiplied by (n - 1). This contradicts our earlier findings where we established that the first term is -7.5 and the common difference is -1.5. The incorrect assignment of these values makes this option inconsistent with the given arithmetic sequence. To further illustrate this, if we substitute n = 1 into this formula, we get a_1 = -1.5 + (-7.5)(1 - 1) = -1.5, which does not match the first term of the sequence (-7.5). This discrepancy confirms that this option is not the correct explicit formula. The misidentification of the first term and the common difference makes this option an unlikely candidate for the correct answer.

Option 4: $a_n=-1.5+a(-7.5-1)$

Similar to Option 2, this option, a_n = -1.5 + a(-7.5 - 1), contains the ambiguous term 'a' and lacks the crucial (n - 1) component. The expression (-7.5 - 1) simplifies to -8.5, which does not correspond to the common difference of -1.5. The presence of 'a' without definition and the absence of the (n - 1) term make this option inconsistent with the general form of an explicit formula for arithmetic sequences. The incorrect numerical values and the structural flaws further reinforce the conclusion that this option is not the correct representation of the sequence. The lack of clarity regarding the 'a' term and the deviation from the standard formulaic structure make this option an unlikely candidate for the correct answer. Therefore, we can confidently eliminate this option from our consideration.

Conclusion: The Correct Explicit Formula

After a thorough analysis of the given options, we can confidently conclude that Option 1, a_n = -7.5 + (-1.5)(n - 1), is the correct explicit formula for the arithmetic sequence -7.5, -9, -10.5, -12, .... This option accurately incorporates the first term (-7.5) and the common difference (-1.5) in the standard format for an explicit formula. The (n - 1) term correctly represents the number of times the common difference is added to the first term to obtain the nth term. In contrast, Options 2, 3, and 4 presented inconsistencies with the fundamental principles of arithmetic sequences and their explicit formulas. Option 2 and 4 contained an ambiguous term 'a' and lacked the crucial (n - 1) component, while Option 3 incorrectly assigned the first term and the common difference. The explicit formula a_n = -7.5 + (-1.5)(n - 1) allows us to efficiently calculate any term in the sequence by simply substituting the value of n. This formula serves as a concise and powerful representation of the arithmetic sequence, enabling us to analyze and understand its behavior. This exercise demonstrates the importance of understanding the structure of arithmetic sequences and their explicit formulas, as well as the ability to carefully analyze and compare different options to arrive at the correct answer.

Final Answer

The explicit formula for the arithmetic sequence $-7.5, -9, -10.5, -12, ext{...}$ is:

$\boxed{a_n=-7.5+(-1.5)(n-1)}$