Eli Runs 2 Miles On His First Day Of Training For A Road Race. The Next Day, He Increases His Distance By 0.25 Miles. Each Subsequent Day, He Continues To Increase His Distance By 0.25 Miles.What Is The Total Number Of Miles Eli Runs Over The Course Of

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Introduction

Eli is a determined individual who has set his sights on completing a road race. To prepare for the event, he has started a training program that involves running a certain distance each day. On his first day of training, Eli runs 2 miles. The next day, he increases his distance by 0.25 miles, and this pattern continues for each subsequent day. In this article, we will explore the concept of a geometric sequence and use it to calculate the total number of miles Eli runs over the course of his training program.

Understanding Geometric Sequences

A geometric sequence is a type of sequence in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In Eli's case, the common ratio is 1.25, since he increases his distance by 0.25 miles each day. The formula for the nth term of a geometric sequence is given by:

an = ar^(n-1)

where an is the nth term, a is the first term, r is the common ratio, and n is the term number.

Calculating the Total Distance

To calculate the total distance Eli runs over the course of his training program, we need to find the sum of the distances he runs each day. Since the distances form a geometric sequence, we can use the formula for the sum of a geometric series to find the total distance.

The formula for the sum of a geometric series is given by:

S = a(1 - r^n) / (1 - r)

where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.

In Eli's case, the first term (a) is 2 miles, the common ratio (r) is 1.25, and we need to find the number of terms (n). Since Eli runs a certain distance each day for a total of 10 days, we can set n = 10.

Plugging in the Values

Now that we have the values for a, r, and n, we can plug them into the formula for the sum of a geometric series:

S = 2(1 - 1.25^10) / (1 - 1.25)

S = 2(1 - 4.322) / (-0.25)

S = 2(-3.322) / (-0.25)

S = 6.644 / (-0.25)

S = -26.576

However, since we are calculating the total distance, which cannot be negative, we need to take the absolute value of the result:

S = |-26.576|

S = 26.576

Conclusion

In conclusion, Eli runs a total of 26.576 miles over the course of his 10-day training program. This is a significant distance, and it is clear that Eli is committed to his training program. By using the concept of a geometric sequence and the formula for the sum of a geometric series, we were able to calculate the total distance Eli runs.

Calculating the Total Distance Using a Formula

Alternatively, we can use a formula to calculate the total distance. The formula for the sum of a geometric series can be rewritten as:

S = a(r^n - 1) / (r - 1)

where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.

In Eli's case, the first term (a) is 2 miles, the common ratio (r) is 1.25, and we need to find the number of terms (n). Since Eli runs a certain distance each day for a total of 10 days, we can set n = 10.

Plugging in the Values

Now that we have the values for a, r, and n, we can plug them into the formula:

S = 2(1.25^10 - 1) / (1.25 - 1)

S = 2(4.322 - 1) / 0.25

S = 2(3.322) / 0.25

S = 6.644 / 0.25

S = 26.576

Conclusion

In conclusion, Eli runs a total of 26.576 miles over the course of his 10-day training program. This is a significant distance, and it is clear that Eli is committed to his training program. By using the concept of a geometric sequence and the formula for the sum of a geometric series, we were able to calculate the total distance Eli runs.

Eli's Road to Success: A Real-World Application

Eli's training program is a real-world application of the concept of a geometric sequence. By increasing his distance by 0.25 miles each day, Eli is creating a geometric sequence that can be used to calculate the total distance he runs. This is a great example of how math can be used to solve real-world problems.

Conclusion

Q: What is a geometric sequence?

A: A geometric sequence is a type of sequence in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: How does Eli's training program relate to a geometric sequence?

A: Eli's training program is a great example of a geometric sequence. Each day, he increases his distance by 0.25 miles, which is the common ratio. This creates a sequence of distances that can be used to calculate the total distance he runs.

Q: What is the formula for the sum of a geometric series?

A: The formula for the sum of a geometric series is given by:

S = a(1 - r^n) / (1 - r)

where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.

Q: How do you calculate the total distance Eli runs using the formula for the sum of a geometric series?

A: To calculate the total distance Eli runs, we need to plug in the values for a, r, and n into the formula. In Eli's case, the first term (a) is 2 miles, the common ratio (r) is 1.25, and we need to find the number of terms (n). Since Eli runs a certain distance each day for a total of 10 days, we can set n = 10.

Q: What is the total distance Eli runs over the course of his 10-day training program?

A: Using the formula for the sum of a geometric series, we can calculate the total distance Eli runs as follows:

S = 2(1 - 1.25^10) / (1 - 1.25)

S = 2(1 - 4.322) / (-0.25)

S = 2(-3.322) / (-0.25)

S = 6.644 / (-0.25)

S = -26.576

However, since we are calculating the total distance, which cannot be negative, we need to take the absolute value of the result:

S = |-26.576|

S = 26.576

Q: What is the significance of Eli's training program?

A: Eli's training program is a great example of how math can be used to solve real-world problems. By increasing his distance by 0.25 miles each day, Eli is creating a geometric sequence that can be used to calculate the total distance he runs. This is a great way to apply mathematical concepts to real-world situations.

Q: How can I use geometric sequences in my own life?

A: Geometric sequences can be used in a variety of real-world situations, such as:

  • Calculating the total distance traveled by a car or other vehicle
  • Determining the amount of money earned by an investment over time
  • Calculating the total cost of a project or other endeavor
  • Determining the amount of time it will take to complete a task or project

Q: What are some common applications of geometric sequences?

A: Geometric sequences have a wide range of applications in fields such as:

  • Finance: Calculating the total value of an investment over time
  • Engineering: Determining the amount of material needed for a project
  • Science: Calculating the total distance traveled by an object
  • Business: Determining the total cost of a project or other endeavor

Q: How can I learn more about geometric sequences?

A: There are many resources available to learn more about geometric sequences, including:

  • Online tutorials and videos
  • Math textbooks and workbooks
  • Online courses and degree programs
  • Math apps and software

Conclusion

In conclusion, Eli's training program is a great example of how math can be used to solve real-world problems. By increasing his distance by 0.25 miles each day, Eli is creating a geometric sequence that can be used to calculate the total distance he runs. This is a great way to apply mathematical concepts to real-world situations.