Using Series To Solve Problems In Mathematics
In this article, we will explore how series can be used to solve real-world problems. We will use a specific example involving Melissa, who is training for a race. By working through this example, we will learn how to represent a problem using summation notation and how to calculate the total distance traveled using the concept of arithmetic series. This problem-solving approach is a fundamental skill in mathematics and can be applied to a variety of scenarios beyond just athletic training.
Melissa has embarked on a training journey to prepare for a race. Her training regimen involves biking, and she follows a specific pattern to gradually increase her endurance. During her initial training session, Melissa bikes a distance of 5 miles. Each subsequent training session, she increases the biking distance by 0.5 miles compared to her previous session. Our goal is to analyze Melissa's training progress using mathematical series. We will represent her training pattern using summation notation and calculate the total distance she covers over a series of training sessions. Understanding this problem not only helps us track Melissa's progress but also provides a framework for solving similar problems involving sequential increments or decrements.
a) Representing Melissa's Training with Summation Notation
To effectively analyze Melissa's training progression, we can use summation notation, a powerful tool in mathematics for representing the sum of a sequence of numbers. Summation notation allows us to express a series in a compact and concise form. In Melissa's case, the distance she bikes in each training session forms an arithmetic sequence, where the difference between consecutive terms is constant (0.5 miles in this scenario). To represent this using summation notation, we first need to identify the general term of the sequence. Let's denote the distance Melissa bikes in the _n_th training session as a_n. Since she bikes 5 miles in the first session and increases the distance by 0.5 miles each time, we can express a_n as:
a_n = 5 + (n - 1) * 0.5
This formula represents an arithmetic sequence with the first term being 5 and a common difference of 0.5. The term (n - 1) accounts for the fact that the first session corresponds to n = 1, and the distance increases by 0.5 miles for each subsequent session. Now, to represent the total distance Melissa bikes over, say, k training sessions, we use summation notation as follows:
∑[n=1 to k] (5 + (n - 1) * 0.5)
This notation reads as "the sum from n = 1 to k of 5 + (n - 1) * 0.5." Here, the Greek letter sigma (∑) denotes summation, n is the index of summation, 1 is the lower limit (starting point), and k is the upper limit (ending point). The expression inside the parentheses, 5 + (n - 1) * 0.5, is the _n_th term of the series. This compact notation efficiently represents the sum of the distances Melissa bikes in each of the k training sessions. By substituting different values of k, we can calculate the total distance for any number of training sessions, making this a versatile tool for analyzing her training progress.
b) Calculating Total Distance After 10 Training Sessions
Now that we have represented Melissa's training progression using summation notation, the next step is to calculate the total distance she bikes after a specific number of training sessions. In this case, we want to determine the total distance after 10 training sessions. This involves evaluating the summation we derived in the previous section for k = 10. The summation notation for the total distance after 10 sessions is:
∑[n=1 to 10] (5 + (n - 1) * 0.5)
To evaluate this sum, we can either manually add up the distances for each of the 10 sessions or use the formula for the sum of an arithmetic series. The formula for the sum of an arithmetic series is:
S_n = (n/2) * (a_1 + a_n)
Where:
- S_n is the sum of the first n terms
- n is the number of terms
- a_1 is the first term
- a_n is the _n_th term
In our case, n = 10, a_1 = 5 (the distance of the first session), and we need to calculate a_10, which is the distance of the 10th session. Using the formula for the _n_th term of an arithmetic sequence, we have:
a_10 = 5 + (10 - 1) * 0.5 = 5 + 9 * 0.5 = 5 + 4.5 = 9.5
So, the distance Melissa bikes in the 10th session is 9.5 miles. Now we can use the formula for the sum of an arithmetic series to find the total distance after 10 sessions:
S_10 = (10/2) * (5 + 9.5) = 5 * 14.5 = 72.5
Therefore, Melissa bikes a total of 72.5 miles after 10 training sessions. This calculation demonstrates how summation notation and the formula for the sum of an arithmetic series can be used to efficiently solve problems involving sequential increments, providing a clear picture of cumulative progress over time. This method is applicable not only to athletic training scenarios but also to various other situations where quantities increase or decrease in a consistent manner.
c) Determining the Number of Sessions to Reach a Total Distance of 100 Miles
Having calculated the total distance Melissa bikes after a fixed number of training sessions, let's now consider a slightly different but related question: How many training sessions does Melissa need to complete to reach a total distance of 100 miles? This problem requires us to work with the same arithmetic series, but instead of finding the sum for a given number of terms, we need to find the number of terms required to reach a specific sum. We will again use the formula for the sum of an arithmetic series, but this time we will solve for n, the number of terms. Recall the formula:
S_n = (n/2) * (a_1 + a_n)
In this case, we know S_n = 100 miles, a_1 = 5 miles, and we need to express a_n in terms of n. We know that a_n = 5 + (n - 1) * 0.5. Substituting this into the sum formula, we get:
100 = (n/2) * (5 + 5 + (n - 1) * 0.5)
Simplifying the equation, we have:
100 = (n/2) * (10 + 0.5n - 0.5) 100 = (n/2) * (9.5 + 0.5n) 200 = n * (9.5 + 0.5n) 200 = 9.5n + 0.5n^2
Rearranging the equation into a quadratic form, we get:
0. 5n^2 + 9.5n - 200 = 0
To solve this quadratic equation, we can either use the quadratic formula or try to factor it. The quadratic formula is given by:
n = [-b ± √(b^2 - 4ac)] / (2a)
Where a = 0.5, b = 9.5, and c = -200. Plugging these values into the formula, we get:
n = [-9.5 ± √(9.5^2 - 4 * 0.5 * (-200))] / (2 * 0.5) n = [-9.5 ± √(90.25 + 400)] / 1 n = [-9.5 ± √490.25] / 1 n = [-9.5 ± 22.14] / 1
We have two possible solutions for n:
n_1 = -9.5 + 22.14 = 12.64 n_2 = -9.5 - 22.14 = -31.64
Since the number of training sessions cannot be negative, we discard the negative solution. Therefore, n ≈ 12.64. Since Melissa cannot complete a fraction of a training session, she needs to complete 13 training sessions to reach a total distance of at least 100 miles. This result demonstrates how we can use mathematical equations, specifically the sum of an arithmetic series and the quadratic formula, to solve real-world problems involving cumulative progress toward a goal. By setting up the equation correctly and solving for the unknown variable, we can make accurate predictions and inform decision-making processes.
In this article, we have explored how series can be used to solve practical problems, using the example of Melissa's training regimen. We learned how to represent her training progression using summation notation, which provides a concise way to express the sum of a sequence of numbers. We then calculated the total distance Melissa bikes after a specific number of training sessions using the formula for the sum of an arithmetic series. Furthermore, we tackled the problem of determining the number of training sessions required to reach a total distance of 100 miles, which involved solving a quadratic equation derived from the sum of the arithmetic series. This process highlights the versatility of mathematical tools in solving real-world problems. The concepts and techniques discussed here, such as summation notation, arithmetic series, and solving equations, are fundamental in mathematics and have wide-ranging applications in various fields beyond just athletic training. Understanding these concepts empowers us to analyze and solve problems involving sequential increments or decrements, making predictions, and making informed decisions based on quantitative analysis. Whether it's tracking progress toward a fitness goal, analyzing financial investments, or modeling population growth, the ability to use series and related mathematical tools is invaluable. The skills acquired through working with series provide a solid foundation for more advanced mathematical concepts and problem-solving techniques, making this a crucial area of study for anyone interested in applying mathematics to real-world scenarios.
