Deriving the Explicit Formula

Oct 13, 2025 by windailys.com 32 views

Introduction

In this article, we will explore the concept of an explicit formula to represent the number of points earned in a video game after a certain number of levels. The problem is as follows: a student starts with 10 points in a video game and earns 7 more points at each level. We will derive an explicit formula to calculate the total number of points earned after n levels.

Understanding the Problem

To start, let's break down the problem and understand the pattern of points earned at each level. The student starts with 10 points and earns 7 more points at each level. This means that at level 1, the student has 10 + 7 = 17 points. At level 2, the student has 17 + 7 = 24 points. At level 3, the student has 24 + 7 = 31 points, and so on.

Identifying the Pattern

From the above examples, we can see that the number of points earned at each level is increasing by 7. This is a classic example of an arithmetic sequence, where each term is obtained by adding a fixed constant to the previous term. In this case, the fixed constant is 7.

Deriving the Explicit Formula

To derive the explicit formula, we need to find a general expression that represents the number of points earned at any level n. Let's denote the number of points earned at level n as P(n). We know that P(1) = 10 + 7 = 17, P(2) = 17 + 7 = 24, and so on. We can see that the number of points earned at each level is increasing by 7, so we can write the following equation:

P(n) = P(n-1) + 7

We can rewrite this equation as:

P(n) = P(1) + (n-1) * 7

Since P(1) = 10, we can substitute this value into the equation:

P(n) = 10 + (n-1) * 7

Simplifying the Formula

We can simplify the formula by evaluating the expression (n-1) * 7:

P(n) = 10 + 7n - 7

Combine like terms:

P(n) = 3 + 7n

Conclusion

In this article, we derived an explicit formula to represent the number of points earned in a video game after n levels. The formula is P(n) = 3 + 7n, where P(n) is the number of points earned at level n. This formula can be used to calculate the total number of points earned at any level n.

Example Use Cases

Here are some example use cases for the formula:

If the student plays for 5 levels, how many points will they have earned? P(5) = 3 + 7(5) = 38
If the student plays for 10 levels, how many points will they have earned? P(10) = 3 + 7(10) = 73
If the student plays for 15 levels, how many points will they have earned? P(15) = 3 + 7(15) = 108

Real-World Applications

The concept of an explicit formula can be applied to a wide range of real-world problems, such as:

Calculating the total cost of a project based on the number of tasks completed
Determining the total amount of money earned by an employee based on their hourly wage and number of hours worked
Calculating the total distance traveled by an object based on its speed and time traveled

Future Research Directions

There are several future research directions that can be explored in this area, such as:

Developing a formula to represent the number of points earned in a video game with multiple levels and different point values
Investigating the use of explicit formulas in other areas of mathematics, such as algebra and calculus
Exploring the application of explicit formulas in real-world problems, such as finance and economics.

Introduction

In our previous article, we derived an explicit formula to represent the number of points earned in a video game after n levels. The formula is P(n) = 3 + 7n, where P(n) is the number of points earned at level n. In this article, we will answer some frequently asked questions about the formula and its applications.

Q: What is the formula for points earned in a video game?

A: The formula for points earned in a video game is P(n) = 3 + 7n, where P(n) is the number of points earned at level n.

Q: How do I use the formula to calculate the number of points earned at a specific level?

A: To use the formula, simply substitute the level number (n) into the equation. For example, if you want to calculate the number of points earned at level 5, you would substitute n = 5 into the equation: P(5) = 3 + 7(5) = 38.

Q: What if I want to calculate the number of points earned at a level that is not a multiple of 7?

A: If you want to calculate the number of points earned at a level that is not a multiple of 7, you can still use the formula. For example, if you want to calculate the number of points earned at level 3, you would substitute n = 3 into the equation: P(3) = 3 + 7(3) = 24.

Q: Can I use the formula to calculate the number of points earned in a video game with multiple levels and different point values?

A: Unfortunately, the formula we derived is only applicable to a video game with a single level and a fixed point value of 7. If you want to calculate the number of points earned in a video game with multiple levels and different point values, you would need to derive a new formula that takes into account the specific point values and level structure of the game.

Q: How can I apply the formula to real-world problems?

A: The concept of an explicit formula can be applied to a wide range of real-world problems, such as calculating the total cost of a project based on the number of tasks completed, determining the total amount of money earned by an employee based on their hourly wage and number of hours worked, and calculating the total distance traveled by an object based on its speed and time traveled.

Q: What are some potential limitations of the formula?

A: One potential limitation of the formula is that it assumes a fixed point value of 7, which may not be applicable to all video games. Additionally, the formula does not take into account any potential bonuses or penalties that may be awarded at specific levels.

Q: Can I use the formula to calculate the number of points earned in a video game with a non-linear level structure?

A: Unfortunately, the formula we derived is only applicable to a video game with a linear level structure. If you want to calculate the number of points earned in a video game with a non-linear level structure, you would need to derive a new formula that takes into account the specific level structure of the game.

Conclusion

In this article, we answered some frequently asked questions about the explicit formula for points earned in a video game. We hope that this article has provided you with a better understanding of the formula and its applications.

Additional Resources

If you want to learn more about explicit formulas and their applications, we recommend checking out the following resources:

Mathematics textbooks: There are many excellent mathematics textbooks that cover the topic of explicit formulas and their applications.
Online resources: There are many online resources available that provide tutorials and examples of explicit formulas and their applications.
Mathematical software: There are many mathematical software packages available that can be used to calculate explicit formulas and their applications.

Future Research Directions

There are several future research directions that can be explored in this area, such as:

Developing a formula to represent the number of points earned in a video game with multiple levels and different point values
Investigating the use of explicit formulas in other areas of mathematics, such as algebra and calculus
Exploring the application of explicit formulas in real-world problems, such as finance and economics