Finding The 10th Term Of The Arithmetic Sequence -4, -1, 2, 5
Let's delve into the world of sequences, specifically arithmetic sequences, and explore how to determine a specific term within them. In this article, we will address the question of finding the 10th term of the arithmetic sequence: $-4, -1, 2, 5, \ldots$ We will break down the concept of arithmetic sequences, identify the pattern in the given sequence, and then apply a formula to efficiently calculate the desired term. By the end of this exploration, you will have a solid understanding of how to tackle similar problems and confidently find terms in arithmetic sequences. So, let's embark on this mathematical journey together!
Understanding Arithmetic Sequences
To effectively tackle the problem, it's crucial to first understand what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference. Let's illustrate this with an example. Consider the sequence: 2, 5, 8, 11, 14,... Notice that the difference between any two consecutive terms is always 3 (5-2=3, 8-5=3, 11-8=3, and so on). Therefore, this is an arithmetic sequence with a common difference of 3. Understanding this fundamental concept is key to solving problems involving arithmetic sequences.
In essence, an arithmetic sequence is a linear progression. Each term is obtained by adding a constant value (the common difference) to the previous term. This makes them predictable and allows us to devise formulas to calculate any term in the sequence without having to list out all the preceding terms. The beauty of arithmetic sequences lies in their simplicity and the patterns they exhibit, making them a cornerstone of many mathematical concepts.
To further solidify your understanding, consider a few more examples. The sequence 1, 4, 7, 10, 13,... is an arithmetic sequence with a common difference of 3. Similarly, the sequence 10, 7, 4, 1, -2,... is also an arithmetic sequence, but this time the common difference is -3 (the terms are decreasing). Recognizing these patterns is the first step towards mastering arithmetic sequences.
Identifying the Common Difference
The common difference is the cornerstone of any arithmetic sequence. To find the common difference (often denoted as 'd'), you simply subtract any term from the term that follows it. For instance, in the sequence 3, 7, 11, 15,... the common difference can be found by subtracting 3 from 7 (7-3=4), or 7 from 11 (11-7=4), and so on. The result will always be the same, confirming that it is indeed an arithmetic sequence. Mastering the identification of the common difference is paramount to unlocking the secrets of arithmetic sequences and solving related problems.
Knowing the common difference allows us to predict the subsequent terms in the sequence. If we know the first term and the common difference, we can construct the entire sequence. This predictability is what makes arithmetic sequences so useful in various mathematical and real-world applications. For example, if we know the rate at which someone is saving money each month (the common difference) and their initial savings, we can predict their total savings after any number of months.
However, finding the common difference sometimes requires careful observation, especially if the sequence involves negative numbers or fractions. For example, consider the sequence: -5, -2, 1, 4,... To find the common difference, we can subtract -5 from -2, which gives us -2 - (-5) = 3. So, the common difference is 3. Paying close attention to signs and performing the subtraction accurately is crucial to avoid errors. Once you've mastered identifying the common difference, you're well on your way to conquering the world of arithmetic sequences.
Analyzing the Given Sequence
Now, let's turn our attention to the sequence provided: $-4, -1, 2, 5, \ldots$ Our first task is to confirm that this is indeed an arithmetic sequence and then determine its common difference. To do this, we'll subtract consecutive terms to see if the difference is constant. Subtracting the first term (-4) from the second term (-1), we get -1 - (-4) = 3. Next, subtracting the second term (-1) from the third term (2), we get 2 - (-1) = 3. Finally, subtracting the third term (2) from the fourth term (5), we get 5 - 2 = 3. Since the difference is consistently 3, we can confidently say that this is an arithmetic sequence with a common difference of 3.
Identifying the common difference is a crucial step in understanding the pattern of the sequence. With the common difference established, we can now predict future terms in the sequence. This ability to forecast terms is one of the key applications of understanding arithmetic sequences. Once you've identified the common difference, you have a powerful tool for analyzing and manipulating the sequence.
Furthermore, by recognizing the first term and the common difference, we can construct a general formula to represent any term in the sequence. This formula will allow us to directly calculate the 10th term, or any other term, without having to manually list out all the preceding terms. The power of this formula lies in its efficiency and its ability to provide a concise representation of the entire sequence. Understanding how to derive and apply this formula is the next step in our journey to solving the problem.
The Formula for the nth Term
The formula for finding the nth term (denoted as a_n) of an arithmetic sequence is a fundamental concept. This formula provides a direct way to calculate any term in the sequence without having to list out all the preceding terms. The formula is given by: $a_n = a_1 + (n - 1)d$ where:
- a_n is the nth term we want to find.
- a_1 is the first term of the sequence.
- n is the position of the term in the sequence (e.g., 10 for the 10th term).
- d is the common difference.
This formula elegantly captures the essence of an arithmetic sequence. It states that any term in the sequence is equal to the first term plus the common difference multiplied by one less than the term's position. The (n-1) factor arises because the first term doesn't have a common difference added to it, the second term has the common difference added once, the third term twice, and so on.
Let's break down the formula with an example. Suppose we have an arithmetic sequence with a first term of 2 and a common difference of 3. To find the 5th term, we would plug the values into the formula: $a_5 = 2 + (5 - 1) * 3 = 2 + 4 * 3 = 2 + 12 = 14$ So, the 5th term of the sequence is 14. Mastering this formula is essential for efficiently solving problems involving arithmetic sequences. Understanding the origin and application of this formula will empower you to tackle a wide range of problems with confidence.
Applying the Formula to Our Problem
Now, let's apply the formula we've learned to the specific problem at hand. We want to find the 10th term of the sequence $-4, -1, 2, 5, \ldots$ We've already established that this is an arithmetic sequence with a common difference (d) of 3. We also know that the first term (a_1) is -4. And we want to find the 10th term, so n = 10. Plugging these values into the formula $a_n = a_1 + (n - 1)d$, we get: $a_{10} = -4 + (10 - 1) * 3$ This is the key step where we translate the problem into a mathematical equation that we can solve. By correctly substituting the values, we set ourselves up for a straightforward calculation that will lead us to the answer. Once the values are plugged in, the problem becomes a simple matter of arithmetic, which we will tackle in the next step. Paying close attention to detail and ensuring accurate substitution is crucial for arriving at the correct answer.
Calculating the 10th Term
With the values substituted into the formula, we now have: $a_10} = -4 + (10 - 1) * 3$ Let's simplify this step-by-step. First, we perform the operation inside the parentheses = -4 + 9 * 3$ Next, we perform the multiplication: 9 * 3 = 27. This gives us: $a_{10} = -4 + 27$ Finally, we perform the addition: -4 + 27 = 23. Therefore, the 10th term of the sequence is 23. This methodical approach to solving the equation ensures accuracy and minimizes the risk of errors. Breaking down the calculation into smaller steps makes it easier to follow and understand the process.
The result, 23, represents the value of the 10th number in the sequence. This means if we were to continue the sequence by adding 3 to each term, the 10th number we would encounter would be 23. This understanding provides a tangible interpretation of the result and reinforces the concept of arithmetic sequences. Now that we have successfully calculated the 10th term, we can confidently choose the correct answer from the given options.
The Answer
Based on our calculations, the 10th term of the sequence $-4, -1, 2, 5, \ldots$ is 23. Therefore, the correct answer is B. 23. We arrived at this answer by first understanding the concept of arithmetic sequences and identifying the common difference. Then, we applied the formula for the nth term of an arithmetic sequence and carefully substituted the known values. Finally, we performed the arithmetic operations to arrive at the solution. This systematic approach highlights the power of mathematical reasoning and problem-solving techniques.
Choosing the correct answer is not just about arriving at the right number; it's about understanding the underlying concepts and the process of problem-solving. By carefully analyzing the sequence, applying the appropriate formula, and performing the calculations accurately, we have demonstrated a solid understanding of arithmetic sequences. This knowledge can be applied to a wide range of mathematical problems and real-world scenarios. Understanding the solution process is as important as getting the right answer.
In conclusion, we have successfully found the 10th term of the given arithmetic sequence by applying the formula and performing the necessary calculations. This process demonstrates the power of mathematical formulas and their ability to efficiently solve problems. By understanding the underlying concepts and applying them systematically, we can confidently tackle similar challenges in the future.