Explicit Formula For The Arithmetic Sequence -42, -30, -18, -6
In the realm of mathematics, sequences play a pivotal role in understanding patterns and relationships between numbers. Among these, arithmetic sequences hold a special significance due to their consistent and predictable nature. An arithmetic sequence is characterized by a constant difference between consecutive terms, a feature that allows us to define them using explicit formulas. In this article, we will delve into the process of deriving the explicit formula for a given arithmetic sequence, providing a comprehensive understanding of the underlying concepts and techniques. Let's embark on this mathematical journey and unlock the secrets of arithmetic sequences.
Decoding the Arithmetic Sequence: -42, -30, -18, -6...
Our mission is to determine the explicit formula for the arithmetic sequence: -42, -30, -18, -6, and so on. To achieve this, we need to identify the key components that define an arithmetic sequence: the first term (A(1)) and the common difference (d). The explicit formula, which we aim to construct, will then express the nth term (A(n)) of the sequence in terms of n, A(1), and d. This formula will serve as a powerful tool for calculating any term in the sequence without having to iterate through all the preceding terms.
Identifying the First Term (A(1))
The first term of a sequence is simply the initial value in the sequence. In our case, the sequence begins with -42. Therefore, we can confidently state that:
A(1) = -42
The first term serves as the foundation upon which the rest of the sequence is built. It is the starting point from which we add the common difference to generate subsequent terms.
Unveiling the Common Difference (d)
The common difference (d) is the constant value that is added to each term to obtain the next term in the sequence. To find the common difference, we can subtract any term from its succeeding term. Let's take the second term (-30) and subtract the first term (-42):
d = -30 - (-42) = -30 + 42 = 12
We can verify this by checking if the same difference holds true for other consecutive terms. Let's subtract the second term (-30) from the third term (-18):
d = -18 - (-30) = -18 + 30 = 12
Indeed, the common difference remains consistent. Therefore, we can conclude that:
d = 12
The common difference is the driving force behind the arithmetic sequence, determining the rate at which the terms increase or decrease.
Constructing the Explicit Formula: A(n) = A(1) + (n - 1)d
Now that we have identified the first term (A(1) = -42) and the common difference (d = 12), we are well-equipped to construct the explicit formula for the sequence. The general formula for the nth term of an arithmetic sequence is given by:
A(n) = A(1) + (n - 1)d
This formula elegantly captures the essence of an arithmetic sequence. It states that the nth term is equal to the sum of the first term and (n - 1) times the common difference. The (n - 1) factor arises because we add the common difference (n - 1) times to the first term to reach the nth term.
Plugging in the Values
To obtain the explicit formula for our specific sequence, we substitute the values of A(1) and d into the general formula:
A(n) = -42 + (n - 1) * 12
Simplifying the Formula
We can further simplify this formula by distributing the 12 and combining like terms:
A(n) = -42 + 12n - 12
A(n) = 12n - 54
Therefore, the explicit formula for the sequence -42, -30, -18, -6... is:
A(n) = 12n - 54
This formula provides a concise and powerful way to calculate any term in the sequence. For instance, to find the 10th term, we simply substitute n = 10 into the formula:
A(10) = 12 * 10 - 54 = 120 - 54 = 66
Summarizing the Key Elements
Let's summarize the key elements we have identified for this arithmetic sequence:
- A(n) = 12n - 54 (Explicit Formula)
- A(1) = -42 (First Term)
- d = 12 (Common Difference)
These three components encapsulate the entire behavior of the arithmetic sequence. The explicit formula allows us to predict any term, the first term anchors the sequence, and the common difference dictates the sequence's progression.
The Power of Explicit Formulas
Explicit formulas are invaluable tools for working with arithmetic sequences. They provide a direct means of calculating any term in the sequence, eliminating the need for recursive calculations. This is particularly useful when dealing with large values of n, where calculating terms iteratively would be cumbersome and time-consuming. Moreover, explicit formulas offer a concise representation of the sequence, highlighting the relationship between the term number (n) and the term value (A(n)).
Conclusion: Mastering Arithmetic Sequences
In this exploration, we have successfully derived the explicit formula for the arithmetic sequence -42, -30, -18, -6.... By identifying the first term and the common difference, we were able to construct the formula A(n) = 12n - 54. This formula empowers us to calculate any term in the sequence with ease and provides a deeper understanding of the sequence's underlying structure. Mastering the concept of explicit formulas is crucial for working with arithmetic sequences and unlocking their mathematical potential. This knowledge serves as a stepping stone for more advanced mathematical concepts and applications.
Through this detailed analysis, we have not only found the explicit formula but also reinforced the fundamental principles of arithmetic sequences. The ability to dissect a sequence, identify its key components, and express it in a concise formula is a testament to the power of mathematical reasoning. As you continue your mathematical journey, remember the principles we have discussed here, and you will be well-equipped to tackle a wide range of sequence-related problems.
The world of mathematics is filled with patterns and relationships, and arithmetic sequences are just one example of the beauty and order that can be found within numbers. By understanding these concepts, we gain a deeper appreciation for the elegance and power of mathematics.
