Explicit Formula And Substitution: Finding The 12th Term Of A Sequence

In the realm of mathematics, particularly when dealing with sequences, explicit formulas serve as powerful tools for defining and understanding patterns. An explicit formula provides a direct way to calculate any term in a sequence without needing to know the preceding terms. This is especially useful in arithmetic sequences, where the difference between consecutive terms remains constant. This constant difference is known as the common difference, often denoted by 'd'. At its core, an arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is constant. This constant difference is called the common difference. Imagine a staircase where each step is the same height; that's essentially an arithmetic sequence in action. Each step represents a term, and the constant height difference is the common difference. Understanding this foundational concept is crucial because it lays the groundwork for the explicit formula, which we'll explore in detail. The beauty of an explicit formula lies in its ability to pinpoint any term directly, regardless of its position in the sequence. This contrasts with recursive formulas, which require knowing the previous term to find the next. The general form of an arithmetic sequence's explicit formula is: an = a1 + (n - 1)d, where an represents the nth term, a1 is the first term, n is the term number, and d is the common difference. This formula is the cornerstone of arithmetic sequence analysis. The common difference (d) is the heartbeat of an arithmetic sequence. It's the constant value added or subtracted to get from one term to the next. To find the common difference, simply subtract any term from its succeeding term. For instance, if our sequence is 2, 5, 8, 11..., the common difference is 5 - 2 = 3. Understanding how to identify the common difference is the key to unlocking the sequence's pattern and, consequently, its explicit formula. Constructing an explicit formula involves identifying the first term (a1) and the common difference (d), then plugging these values into the general formula. Suppose we have an arithmetic sequence starting with 7, and the common difference is 4. Our explicit formula would be: an = 7 + (n - 1)4. This formula now allows us to calculate any term in the sequence with ease.

Let's delve into the explicit formula provided: aₙ = 18 + 2(n-1). Our main keyword here is the explicit formula, which is a mathematical expression that directly calculates the nth term of a sequence. By dissecting this formula, we can extract valuable information about the arithmetic sequence it represents. The formula aₙ = 18 + 2(n-1) is a compact yet powerful representation of an arithmetic sequence. It tells us exactly how to find any term (aₙ) in the sequence based solely on its position (n). No need to calculate previous terms; just plug in the value of 'n', and the formula delivers the result. Think of it as a mathematical shortcut, bypassing the step-by-step addition of the common difference. The number 18 in the formula holds significant meaning. It represents the first term (a₁) of the arithmetic sequence. This is the starting point, the foundation upon which the rest of the sequence is built. Without the first term, we wouldn't know where the sequence begins. The presence of 18 anchors the sequence and provides a reference point for all subsequent terms. This is a crucial piece of information, much like knowing the starting line in a race. The number 2, sitting outside the parentheses, reveals the common difference (d) of our sequence. As we've established, the common difference is the constant value added to each term to obtain the next. In this case, each term increases by 2. This steady progression is the hallmark of an arithmetic sequence, and the number 2 encapsulates this consistent growth. It's like the consistent stride length of a runner, dictating the pace of the sequence. The term (n-1) within the formula plays a vital role in determining how many times the common difference is added. Since we start with the first term (a₁), we don't add the common difference to it. For the second term (a₂), we add the common difference once; for the third term (a₃), we add it twice, and so on. The (n-1) term elegantly captures this pattern, ensuring that we add the common difference the correct number of times. It's a subtle but essential component of the formula, ensuring accuracy as we move through the sequence. When we expand the formula aₙ = 18 + 2(n-1), we're essentially distributing the 2 and simplifying the expression. This can give us a different perspective on the sequence. Distributing the 2, we get aₙ = 18 + 2n - 2. Combining like terms, we arrive at aₙ = 2n + 16. This simplified form highlights the linear nature of arithmetic sequences, where the nth term is a linear function of n. The coefficient of n (which is 2 in this case) is the common difference, and the constant term (16) is related to the first term.

Now, let's shift our focus to the practical application of the formula. Our goal is to substitute n = 12 into the formula aₙ = 18 + 2(n-1). This process will allow us to pinpoint the 12th term (a₁₂) of the sequence. Substitution is a fundamental operation in mathematics, allowing us to evaluate expressions and equations for specific values. In this context, we're replacing the variable 'n' with the number 12, effectively telling the formula to calculate the term in the 12th position. It's like selecting a specific page in a book; we're directing the formula to a particular location within the sequence. When we substitute n = 12 into the formula aₙ = 18 + 2(n-1), we're initiating a series of calculations. First, we replace 'n' with 12, resulting in a₁₂ = 18 + 2(12-1). This sets the stage for the arithmetic operations that will reveal the 12th term. The substitution step is crucial; it transforms the general formula into a specific calculation tailored to our desired term. Following the order of operations (PEMDAS/BODMAS), we first tackle the expression within the parentheses. In our case, this means calculating 12 - 1, which equals 11. This step simplifies the formula to a₁₂ = 18 + 2(11). Parentheses often indicate a grouping or a priority in calculations, so addressing them first is essential for accuracy. Next in line is multiplication. We multiply 2 by 11, which gives us 22. Our formula now looks like this: a₁₂ = 18 + 22. Multiplication is a fundamental arithmetic operation, and in this context, it represents adding the common difference (2) eleven times to the first term (18). Finally, we perform the addition. Adding 18 and 22, we arrive at 40. Therefore, a₁₂ = 40. This is the culmination of our calculation, the answer we've been seeking. The 12th term of the arithmetic sequence defined by the formula aₙ = 18 + 2(n-1) is 40. This result demonstrates the power and efficiency of explicit formulas in determining specific terms within a sequence. Our final answer, a₁₂ = 40, signifies that the 12th number in our sequence is 40. This is not just a numerical value; it's a data point within a larger pattern. It tells us where the sequence is heading and provides a snapshot of its growth. Understanding how to arrive at this answer, through substitution and arithmetic operations, is the key takeaway.

In conclusion, the explicit formula is a cornerstone of understanding and working with arithmetic sequences. It allows us to calculate any term directly, providing a powerful tool for mathematical analysis and problem-solving. By dissecting the formula and performing the substitution, we've successfully identified the 12th term of the sequence, further solidifying our understanding of these concepts. The journey through explicit formulas and substitutions unveils the beauty and efficiency of mathematical tools. We've seen how a single formula can encapsulate an entire sequence and how substitution allows us to pinpoint specific terms with ease. This mastery not only solves individual problems but also strengthens our foundation for more advanced mathematical explorations. The significance of explicit formulas extends far beyond textbook exercises. They are the building blocks for more complex mathematical models and algorithms used in various fields, from computer science to finance. Understanding these formulas empowers us to analyze patterns, predict outcomes, and solve real-world problems with precision.