For A Fund-raiser, Anthony Earns $5 For The First Mile He Runs, An Additional $7 For The Second Mile, An Additional $9 For The Third Mile, And So On. The Expression $\sum_{n=1}^{10}[5+2(n-1)] Represents The Total Amount Of Money

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Introduction

Anthony is participating in a fund-raiser where he earns money based on the distance he runs. The amount he earns is determined by a specific expression that takes into account the number of miles he runs. In this article, we will delve into the expression ∑n=110[5+2(n−1)]\sum_{n=1}^{10}[5+2(n-1)] and understand how it represents the total amount of money Anthony earns.

The Expression: A Breakdown

The expression ∑n=110[5+2(n−1)]\sum_{n=1}^{10}[5+2(n-1)] can be broken down into several components. Let's start by understanding the individual parts of the expression.

  • Summation notation: The expression is written in summation notation, which is a compact way of representing a sum of terms. The summation notation is represented by the symbol ∑\sum, and it indicates that we need to sum up the terms inside the brackets.
  • Expression inside the brackets: The expression inside the brackets is 5+2(n−1)5+2(n-1). This expression represents the amount of money Anthony earns for each mile he runs.
  • Index of summation: The index of summation is nn, which represents the number of miles Anthony runs. The index starts from 1 and goes up to 10, indicating that Anthony runs a total of 10 miles.

Understanding the Expression

Now that we have broken down the expression, let's understand how it represents the total amount of money Anthony earns. The expression can be expanded as follows:

∑n=110[5+2(n−1)]=(5+2(1−1))+(5+2(2−1))+(5+2(3−1))+…+(5+2(10−1))\sum_{n=1}^{10}[5+2(n-1)] = (5+2(1-1)) + (5+2(2-1)) + (5+2(3-1)) + \ldots + (5+2(10-1))

Simplifying the expression, we get:

∑n=110[5+2(n−1)]=5+5+5+…+5+2(9)+2(8)+…+2(1)\sum_{n=1}^{10}[5+2(n-1)] = 5 + 5 + 5 + \ldots + 5 + 2(9) + 2(8) + \ldots + 2(1)

This expression represents the total amount of money Anthony earns, which is the sum of the amounts he earns for each mile he runs.

Calculating the Total Amount

To calculate the total amount of money Anthony earns, we need to evaluate the expression. Let's start by evaluating the terms inside the brackets:

  • (5+2(1−1))=5(5+2(1-1)) = 5
  • (5+2(2−1))=7(5+2(2-1)) = 7
  • (5+2(3−1))=9(5+2(3-1)) = 9
  • …\ldots
  • (5+2(10−1))=23(5+2(10-1)) = 23

Now, let's add up the terms:

5+7+9+…+23=1455 + 7 + 9 + \ldots + 23 = 145

Therefore, the total amount of money Anthony earns is $145\$145.

Conclusion

In this article, we have discussed Anthony's earning expression and how it represents the total amount of money he earns for participating in a fund-raiser. We have broken down the expression into its individual components and evaluated it to find the total amount of money Anthony earns. The expression ∑n=110[5+2(n−1)]\sum_{n=1}^{10}[5+2(n-1)] is a powerful tool for calculating the total amount of money Anthony earns, and it can be used to understand the relationship between the number of miles he runs and the amount of money he earns.

Understanding the Formula

The expression ∑n=110[5+2(n−1)]\sum_{n=1}^{10}[5+2(n-1)] can be represented by a formula:

∑n=110[5+2(n−1)]=n(n+1)2⋅5+2⋅n(n−1)2\sum_{n=1}^{10}[5+2(n-1)] = \frac{n(n+1)}{2} \cdot 5 + 2 \cdot \frac{n(n-1)}{2}

This formula can be used to calculate the total amount of money Anthony earns for any number of miles he runs.

Derivation of the Formula

To derive the formula, we can start by evaluating the expression:

∑n=110[5+2(n−1)]=(5+2(1−1))+(5+2(2−1))+(5+2(3−1))+…+(5+2(10−1))\sum_{n=1}^{10}[5+2(n-1)] = (5+2(1-1)) + (5+2(2-1)) + (5+2(3-1)) + \ldots + (5+2(10-1))

Simplifying the expression, we get:

∑n=110[5+2(n−1)]=5+5+5+…+5+2(9)+2(8)+…+2(1)\sum_{n=1}^{10}[5+2(n-1)] = 5 + 5 + 5 + \ldots + 5 + 2(9) + 2(8) + \ldots + 2(1)

This expression can be represented by a formula:

∑n=110[5+2(n−1)]=n(n+1)2⋅5+2⋅n(n−1)2\sum_{n=1}^{10}[5+2(n-1)] = \frac{n(n+1)}{2} \cdot 5 + 2 \cdot \frac{n(n-1)}{2}

This formula can be used to calculate the total amount of money Anthony earns for any number of miles he runs.

Real-World Applications

The expression ∑n=110[5+2(n−1)]\sum_{n=1}^{10}[5+2(n-1)] has several real-world applications. For example:

  • Fund-raising events: The expression can be used to calculate the total amount of money earned by participants in a fund-raising event.
  • Sales and marketing: The expression can be used to calculate the total amount of money earned by sales and marketing teams.
  • Financial planning: The expression can be used to calculate the total amount of money earned by individuals or businesses.

Conclusion

Introduction

In our previous article, we discussed Anthony's earning expression and how it represents the total amount of money he earns for participating in a fund-raiser. We broke down the expression into its individual components and evaluated it to find the total amount of money Anthony earns. In this article, we will answer some frequently asked questions about Anthony's earning expression.

Q: What is the formula for Anthony's earning expression?

A: The formula for Anthony's earning expression is:

∑n=110[5+2(n−1)]=n(n+1)2⋅5+2⋅n(n−1)2\sum_{n=1}^{10}[5+2(n-1)] = \frac{n(n+1)}{2} \cdot 5 + 2 \cdot \frac{n(n-1)}{2}

Q: How does the formula work?

A: The formula works by first calculating the sum of the amounts earned for each mile run, and then adding up the amounts earned for each mile. The formula takes into account the number of miles run and the amount earned for each mile.

Q: What is the significance of the index of summation?

A: The index of summation, n, represents the number of miles run. The index starts from 1 and goes up to 10, indicating that Anthony runs a total of 10 miles.

Q: How can I use Anthony's earning expression in real-world applications?

A: Anthony's earning expression can be used in various real-world applications, such as:

  • Fund-raising events: The expression can be used to calculate the total amount of money earned by participants in a fund-raising event.
  • Sales and marketing: The expression can be used to calculate the total amount of money earned by sales and marketing teams.
  • Financial planning: The expression can be used to calculate the total amount of money earned by individuals or businesses.

Q: What are some common mistakes to avoid when using Anthony's earning expression?

A: Some common mistakes to avoid when using Anthony's earning expression include:

  • Not understanding the formula: Make sure you understand how the formula works and what it represents.
  • Not using the correct index of summation: Make sure you use the correct index of summation, n, to represent the number of miles run.
  • Not taking into account the number of miles run: Make sure you take into account the number of miles run when using the expression.

Q: How can I modify Anthony's earning expression to suit my needs?

A: You can modify Anthony's earning expression to suit your needs by:

  • Changing the amount earned for each mile: You can change the amount earned for each mile by modifying the formula.
  • Changing the number of miles run: You can change the number of miles run by modifying the index of summation.
  • Adding or subtracting terms: You can add or subtract terms from the expression to suit your needs.

Conclusion

In this article, we have answered some frequently asked questions about Anthony's earning expression. We have discussed the formula, how it works, and how it can be used in real-world applications. We have also highlighted some common mistakes to avoid and provided tips on how to modify the expression to suit your needs.