Arithmetic Sequence 13, 8, 3, -2, -7, -12 Finding The Nth Term
This article delves into the intriguing world of arithmetic sequences, focusing on the specific sequence 13, 8, 3, -2, -7, -12, .... We will meticulously analyze this sequence to determine its underlying pattern, derive a general formula for the nth term, and explore practical applications of this formula. Whether you're a student grappling with arithmetic sequences or simply someone with a penchant for mathematical patterns, this comprehensive guide will illuminate the intricacies of this fascinating sequence. Our primary objective is to provide a clear, concise, and easily understandable explanation, empowering you to confidently tackle similar problems. Let's embark on this mathematical journey together and unlock the secrets hidden within this arithmetic sequence.
Identifying the Arithmetic Nature of the Sequence
To begin our exploration, we must first establish that the given sequence, 13, 8, 3, -2, -7, -12, ..., is indeed an arithmetic sequence. An arithmetic sequence, by definition, is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference. In our sequence, we can observe the following:
- 8 - 13 = -5
- 3 - 8 = -5
- -2 - 3 = -5
- -7 - (-2) = -5
- -12 - (-7) = -5
The consistent difference of -5 between consecutive terms definitively confirms that this sequence is an arithmetic sequence. This is a crucial first step, as it allows us to apply the well-established formulas and properties associated with arithmetic sequences. Understanding this foundational aspect is essential for navigating the subsequent steps of our analysis. The common difference, -5, plays a pivotal role in determining the general formula and predicting future terms in the sequence. Therefore, correctly identifying the sequence as arithmetic is paramount to our overall understanding and solution.
Deriving the General Formula for the Nth Term
Now that we have confirmed the arithmetic nature of the sequence 13, 8, 3, -2, -7, -12, ..., our next crucial step is to derive the general formula for the nth term. This formula will empower us to calculate any term in the sequence without having to explicitly list out all the preceding terms. The general formula for the nth term (an) of an arithmetic sequence is given by:
an = a1 + (n - 1)d
where:
- an represents the nth term of the sequence
- a1 represents the first term of the sequence
- n represents the term number (e.g., 1 for the first term, 2 for the second term, and so on)
- d represents the common difference between consecutive terms
In our specific sequence, we have:
- a1 = 13 (the first term)
- d = -5 (the common difference, as established earlier)
Substituting these values into the general formula, we get:
an = 13 + (n - 1)(-5)
This formula is the cornerstone of our ability to work with this sequence. It allows us to find any term we desire, simply by substituting the appropriate value of 'n'. Simplifying this formula further will make it even easier to use.
Let's simplify the formula:
an = 13 - 5(n - 1)
This simplified formula, an = 13 - 5(n - 1), provides a concise and efficient way to calculate any term in the sequence 13, 8, 3, -2, -7, -12, .... It encapsulates the inherent pattern of the sequence and allows us to predict its behavior accurately.
Calculating a Specific Term Using the Formula
The power of the general formula an = 13 - 5(n - 1) lies in its ability to efficiently calculate any term in the sequence. Let's illustrate this by calculating a specific term. Suppose we want to find the 10th term (a10) of the sequence. To do this, we simply substitute n = 10 into the formula:
a10 = 13 - 5(10 - 1)
Now, we perform the arithmetic operations:
a10 = 13 - 5(9) a10 = 13 - 45 a10 = -32
Therefore, the 10th term of the sequence is -32. This demonstrates the practical utility of the general formula. We can quickly determine the value of any term without having to manually extend the sequence. This is particularly advantageous when dealing with large term numbers, where listing out all the preceding terms would be tedious and time-consuming. The formula provides a direct and accurate means of accessing any point in the sequence.
Let's consider another example. What if we wanted to find the 25th term (a25)? We would simply substitute n = 25 into the formula:
a25 = 13 - 5(25 - 1) a25 = 13 - 5(24) a25 = 13 - 120 a25 = -107
Thus, the 25th term of the sequence is -107. These examples underscore the efficiency and versatility of the general formula in calculating specific terms within the arithmetic sequence.
Solving for 'n' Given a Term Value
Beyond calculating specific terms, the general formula an = 13 - 5(n - 1) can also be used to solve for 'n', the term number, when a specific term value is given. This is a valuable application of the formula, allowing us to determine the position of a particular value within the sequence. For instance, suppose we are given that a certain term in the sequence has a value of -52, and we want to find out which term it is (i.e., find the value of 'n').
We start by setting an equal to -52 in our formula:
-52 = 13 - 5(n - 1)
Now, we solve for 'n' using algebraic manipulation:
-
Subtract 13 from both sides:
-52 - 13 = -5(n - 1) -65 = -5(n - 1)
-
Divide both sides by -5:
-65 / -5 = n - 1 13 = n - 1
-
Add 1 to both sides:
13 + 1 = n 14 = n
Therefore, the term with a value of -52 is the 14th term in the sequence. This demonstrates how the general formula can be rearranged and utilized to solve for 'n' when the term value is known. This is a powerful technique for analyzing and understanding the structure of arithmetic sequences. It allows us to not only predict term values but also to locate specific values within the sequence.
Practical Applications of Arithmetic Sequences
Arithmetic sequences are not merely abstract mathematical concepts; they have numerous practical applications in various fields. Understanding arithmetic sequences can be beneficial in real-world scenarios, ranging from finance to physics. One common application is in calculating simple interest. The interest earned each year on a fixed principal amount forms an arithmetic sequence. For example, if you deposit $1000 in a bank account with a simple interest rate of 5% per year, the interest earned each year ($50) forms an arithmetic sequence. The total amount in your account each year also forms an arithmetic sequence.
Another application lies in uniformly accelerated motion in physics. The distance traveled by an object with constant acceleration in equal intervals of time forms an arithmetic sequence. For instance, a ball rolling down an inclined plane experiences constant acceleration due to gravity. The distances it covers in each second will form an arithmetic sequence. This principle is fundamental in understanding and predicting the motion of objects under constant acceleration.
Arithmetic sequences are also used in computer programming and data analysis. They can be used to generate sequences of numbers for various purposes, such as indexing data or creating patterns. In spreadsheet software, arithmetic sequences can be easily generated using formulas, making it simple to create tables of values or analyze trends. The ability to model and predict patterns using arithmetic sequences is a valuable skill in many technical fields.
Furthermore, arithmetic sequences find applications in everyday situations. Consider a seating arrangement in a theater where each row has a fixed number of seats more than the previous row. The number of seats in each row forms an arithmetic sequence. Similarly, the cost of renting equipment, where there is a fixed daily rate plus an initial fee, can be modeled using an arithmetic sequence. These examples illustrate the pervasive nature of arithmetic sequences and their relevance in diverse contexts.
Conclusion: Mastering Arithmetic Sequences
In conclusion, our exploration of the arithmetic sequence 13, 8, 3, -2, -7, -12, ... has provided a comprehensive understanding of its properties and applications. We began by identifying the sequence as arithmetic, establishing the crucial common difference of -5. This foundation allowed us to derive the general formula for the nth term: an = 13 - 5(n - 1). This formula is a powerful tool, enabling us to calculate any term in the sequence with ease. We demonstrated this by calculating the 10th and 25th terms, showcasing the formula's efficiency and accuracy.
Furthermore, we explored the versatility of the general formula by using it to solve for 'n', the term number, when a specific term value is given. This technique allows us to locate a particular value within the sequence, providing a deeper understanding of its structure. We illustrated this with an example, finding the position of the term with a value of -52. Understanding how to manipulate the formula to solve for different variables is key to mastering arithmetic sequences.
Finally, we discussed the practical applications of arithmetic sequences in various fields, from finance and physics to computer programming and everyday situations. These examples highlight the real-world relevance of this mathematical concept and demonstrate its usefulness in solving practical problems. From calculating simple interest to modeling uniformly accelerated motion, arithmetic sequences provide a valuable framework for understanding and predicting patterns.
By mastering the concepts and techniques presented in this article, you are well-equipped to tackle arithmetic sequences with confidence. The ability to identify arithmetic sequences, derive their general formulas, and apply these formulas to solve problems is a valuable skill in mathematics and beyond. Whether you are a student learning about sequences or someone seeking to enhance your problem-solving abilities, the knowledge gained here will undoubtedly prove beneficial.
