Calculating Hiking Distance Using Arithmetic Sequences
In this article, we will explore the total number of miles Jeremy walked on a hike, which can be represented using the formula a_n = a_{n-1} + 3. Given that a_1 = 2 and n represents the total number of hours walked, we aim to determine the total distance Jeremy walked at the end of a specified time. This problem delves into the realm of arithmetic sequences, a fundamental concept in mathematics. We will dissect the formula, understand its components, and apply it to calculate Jeremy's hiking distance for various values of n. This exploration will not only provide a solution to the specific problem but also enhance your understanding of arithmetic sequences and their applications in real-world scenarios.
The formula a_n = a_n-1} + 3** defines a recursive arithmetic sequence. Let's break down each part represents the previous term in the sequence, meaning the distance Jeremy walked after n-1 hours. The constant 3 is the common difference between consecutive terms, indicating that Jeremy walks an additional 3 miles every hour. This constant addition is the hallmark of an arithmetic sequence.
The initial condition a_1 = 2 is crucial. It tells us that at the end of the first hour (n=1), Jeremy had walked 2 miles. This serves as the starting point for our sequence. To find the distance Jeremy walked at the end of any subsequent hour, we simply add the common difference (3 miles) to the distance he walked in the previous hour. For example, to find a_2, the distance Jeremy walked after 2 hours, we substitute into the formula: a_2 = a_1 + 3 = 2 + 3 = 5 miles. Similarly, a_3 = a_2 + 3 = 5 + 3 = 8 miles, and so on. Understanding this recursive process is key to solving the problem and grasping the concept of arithmetic sequences.
To further illustrate the concept, let's consider a few more hours. After 4 hours, Jeremy would have walked a_4 = a_3 + 3 = 8 + 3 = 11 miles. After 5 hours, a_5 = a_4 + 3 = 11 + 3 = 14 miles. By observing these initial terms, we can see a pattern emerging. The sequence of distances Jeremy walked forms an arithmetic progression: 2, 5, 8, 11, 14, and so on. Each term is obtained by adding 3 to the previous term. This pattern allows us to predict the distance Jeremy walked after any number of hours, provided we continue the sequence. However, for larger values of n, repeatedly applying the recursive formula can become tedious. Therefore, it is beneficial to explore a more direct, explicit formula for this arithmetic sequence.
To determine the total distance Jeremy walked after n hours, we can continue using the recursive formula, but a more efficient approach is to derive an explicit formula for the arithmetic sequence. An explicit formula allows us to directly calculate the nth term (a_n) without having to calculate all the preceding terms. This is particularly useful when we need to find the distance walked after a large number of hours. The general form of an explicit formula for an arithmetic sequence is a_n = a_1 + (n - 1)d, where a_1 is the first term, n is the term number (number of hours in this case), and d is the common difference.
In our problem, we know that a_1 = 2 (the initial distance) and d = 3 (the common difference). Substituting these values into the general formula, we get: a_n = 2 + (n - 1)3. This is the explicit formula for the distance Jeremy walked after n hours. We can simplify this formula further by distributing the 3 and combining like terms: a_n = 2 + 3n - 3 = 3n - 1. Now we have a concise and powerful formula that directly relates the number of hours walked (n) to the total distance (a_n).
Using this explicit formula, we can easily calculate the distance Jeremy walked after any number of hours. For example, if we want to find the distance after 10 hours, we simply substitute n = 10 into the formula: a_10 = 3(10) - 1 = 30 - 1 = 29 miles. This is much faster than repeatedly applying the recursive formula. Similarly, we can find the distance after 20 hours: a_20 = 3(20) - 1 = 60 - 1 = 59 miles. The explicit formula provides a direct and efficient way to calculate the total distance walked for any given number of hours.
Let's consider another example. Suppose we want to know how far Jeremy walked after 24 hours. Using our explicit formula, we calculate a_24 = 3(24) - 1 = 72 - 1 = 71 miles. This demonstrates the power of the explicit formula in quickly determining the distance without having to iterate through each preceding hour. The formula a_n = 3n - 1 encapsulates the entire arithmetic sequence and allows us to easily solve for any term. This underscores the importance of understanding and deriving explicit formulas for arithmetic sequences, especially in problems involving repeated additions or subtractions.
Now that we have both the recursive and explicit formulas for Jeremy's hiking distance, let's explore how to apply them to solve various problems. The recursive formula, a_n = a_{n-1} + 3, is useful for understanding the iterative nature of the sequence and calculating consecutive terms. However, the explicit formula, a_n = 3n - 1, is far more efficient for finding the distance Jeremy walked after a specific number of hours without needing to calculate the preceding distances.
Suppose we want to find out how many hours Jeremy needs to walk to cover a total distance of 100 miles. We can use the explicit formula to solve this problem. We set a_n = 100 and solve for n: 100 = 3n - 1. Adding 1 to both sides gives 101 = 3n. Dividing both sides by 3, we get n = 101/3 ≈ 33.67 hours. Since Jeremy can only walk for a whole number of hours, he would need to walk for 34 hours to cover at least 100 miles. After 33 hours, he would have walked a_33 = 3(33) - 1 = 98 miles, which is less than 100 miles. Therefore, after 34 hours, he would have walked a_34 = 3(34) - 1 = 101 miles, exceeding the 100-mile mark.
Another type of problem we can solve is determining the difference in distance walked between two specific hours. For instance, what is the difference in distance between Jeremy's walk after 15 hours and after 8 hours? Using the explicit formula, we can calculate a_15 = 3(15) - 1 = 44 miles and a_8 = 3(8) - 1 = 23 miles. The difference in distance is 44 - 23 = 21 miles. This illustrates how the explicit formula can be used to quickly compare distances at different points in time.
We can also use the formulas to analyze patterns in Jeremy's hiking distance. For example, how much further does Jeremy walk every 10 hours? Let's compare the distance after 10 hours (a_10 = 3(10) - 1 = 29 miles) to the distance after 20 hours (a_20 = 3(20) - 1 = 59 miles). The difference is 59 - 29 = 30 miles. This is exactly 10 times the common difference (3 miles), which makes sense because Jeremy walks an additional 3 miles each hour. This type of analysis reinforces the connection between the common difference and the linear growth of the arithmetic sequence.
The problem of Jeremy's hiking distance provides a tangible example of arithmetic sequences, but these sequences have numerous applications in various real-world scenarios. Understanding arithmetic sequences can help you model and solve problems in fields ranging from finance to physics.
In the realm of finance, arithmetic sequences can be used to model simple interest calculations. Simple interest is calculated only on the principal amount, and the interest earned each year is constant. For example, if you deposit $1000 in an account that earns $50 in simple interest each year, the total amount in the account each year forms an arithmetic sequence: $1050, $1100, $1150, and so on. The common difference in this sequence is $50, the annual interest earned. Using the formulas for arithmetic sequences, you can easily calculate the total amount in the account after any number of years.
Another financial application is in modeling loan repayments. If you are paying off a loan with fixed monthly payments, the remaining balance on the loan each month forms an arithmetic sequence. The common difference is the amount of the monthly payment, and the initial term is the original loan amount. By understanding this sequence, you can track the progress of your loan repayment and determine when the loan will be fully paid off.
In physics, arithmetic sequences can be used to model the motion of an object with constant acceleration. For example, if an object starts from rest and accelerates at a constant rate, the distance traveled by the object in each successive time interval forms an arithmetic sequence. The common difference depends on the acceleration and the time interval. This application highlights the connection between mathematics and the physical world.
Arithmetic sequences also appear in everyday situations. Consider a stack of chairs where each row has a fixed number of additional chairs compared to the previous row. The number of chairs in each row forms an arithmetic sequence. Similarly, the number of seats in each row of a theater or stadium might follow an arithmetic progression. These examples demonstrate the pervasive nature of arithmetic sequences in our daily lives.
In conclusion, by analyzing Jeremy's hiking distance using the formula a_n = a_{n-1} + 3, we have gained a deeper understanding of arithmetic sequences and their applications. We have explored both the recursive and explicit formulas, learned how to apply them to solve problems, and discovered the relevance of arithmetic sequences in real-world contexts. The explicit formula, a_n = 3n - 1, provides a powerful tool for efficiently calculating the total distance Jeremy walked after any number of hours.
Understanding arithmetic sequences is a fundamental skill in mathematics, and the ability to apply these concepts to practical problems is invaluable. From calculating financial growth to modeling physical phenomena, arithmetic sequences provide a framework for understanding patterns and making predictions. By mastering the concepts discussed in this article, you will be well-equipped to tackle a wide range of mathematical challenges and appreciate the elegance and utility of arithmetic sequences.
The problem of Jeremy's hike serves as an excellent starting point for further exploration of mathematical sequences and series. Beyond arithmetic sequences, there are geometric sequences, Fibonacci sequences, and many other fascinating patterns to discover. By continuing your mathematical journey, you will unlock new insights and develop a deeper appreciation for the power and beauty of mathematics.
