Arithmetic and Geometric Series (tortoise and the hare)

The conversation discusses two formulas for calculating the time it takes for a tortoise and a hare to complete a race, as well as the fact that the tortoise will finish in 100 minutes while the hare may never finish. The work done to solve for the hare's finishing time is shown, but it is concluded that the hare will eventually finish the race but it will take an infinite amount of time. In summary, the conversation discusses the formulas for calculating race completion time and concludes that while the tortoise will finish in 100 minutes, the hare may take an infinite amount of time to finish.
  • #1
pbonnie
92
0

Homework Statement


The question is attachedk


Homework Equations


Sn = n/2[2a+(n-1)d]
Sn = (a x (1-r^n))/1-r


The Attempt at a Solution


I already found the general formulas:
Tortoise:
Sn = n/2(40)
Hare:
Sn = (1000 x [1-0.5^n])/0.5

And I know that there tortoise will finish the race in 100 minutes. But I don't think the hare ever finishes the race? This is the work I did so far trying to solve for the hare:
2000 = (1000(1-0.5^n))/0.5
1000 = 1000(1-0.5^n)
1 = 1-0.5^n
2 = 0.5^n
But 0.5 the the power of anything can never equal 2?

Thank you!
 
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  • #3
Correction* 0 = 0.5^n
Is that the solution? The hare can never finish the race because no matter what value n is, it can never equal 0?
 
  • #4
Correction: it will eventually finish, but takes an infinite amount of time
 
  • #5
eddybob123 said:
Correction: it will eventually finish, but takes an infinite amount of time
Isn't that a contradiction of terms?
 

Related to Arithmetic and Geometric Series (tortoise and the hare)

1. What is the difference between an arithmetic and geometric series?

An arithmetic series is a sequence of numbers where each term is obtained by adding a constant value to the previous term. A geometric series is a sequence of numbers where each term is obtained by multiplying the previous term by a constant value. The main difference between the two is the method of obtaining each term.

2. How do you determine the sum of an arithmetic series?

To find the sum of an arithmetic series, you can use the formula S = (n/2)(a + l), where S is the sum, n is the number of terms, a is the first term, and l is the last term. Another way is to use the formula S = (n/2)(2a + (n-1)d), where d is the common difference between each term.

3. How do you determine the sum of a geometric series?

To find the sum of a geometric series, you can use the formula S = a(1-r^n)/(1-r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. Another way is to use the formula S = (a(1-r^n))/(1-r), where d is the common difference between each term.

4. What is the significance of the "tortoise and the hare" in arithmetic and geometric series?

The "tortoise and the hare" is a popular analogy used to explain the concept of convergence in arithmetic and geometric series. The tortoise represents the arithmetic series, with its steady and predictable pace, while the hare represents the geometric series, with its rapid growth. The analogy shows that even though the hare may start off faster, the tortoise will eventually catch up and surpass the hare due to its steady pace.

5. How are arithmetic and geometric series used in real-world applications?

Arithmetic and geometric series are used in various real-world applications, such as calculating interest rates and loan payments in finance, predicting population growth and decay in biology, and analyzing stock market trends in economics. They are also used in computer science for algorithms and data compression, and in physics for solving problems involving acceleration and velocity.

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