Quantitative Reasoning: Modeling with Sequences and Series (assignment # 2)~ amdm

The more disks that are added to the tower, the more difficult the game becomes. What is the minimum number of moves it would take to move two disks from one peg to the other?

3

The simplest version of the game has only one disk. What is the minimum number of moves it would take to move one disk from one peg to the other?

1

1. The sequence of the differences, 2, 4, 8, 16, 32, ... is a geometric sequence with: an = and r = 2. Which formula can be used to find any term for this sequence? Check all that apply.

1. 2,2 2. an = 2(2)n-1

1. Explain the steps for moving three disks from one peg to another. Start with moving the small disk to the right peg. The least number of moves is

1. What to include: Which steps did you include in your explanation after moving the small disk to the right peg? Move the medium disk to the left peg. Move the small disk to the left peg. Move the large disk to the right peg. Move the small disk to the middle peg. Move the middle disk to the right peg. Move the small disk to the right peg. 2. 7

The sequences are: Number of moves: 1, 3, 7, 15, 31, 63, ... es002-1.jpg: 2, 4, 8, 16, 32, 64, ... Compare the two sequences. What is the pattern? How could you find the minimum number of moves required for a given number of disks?

What to include: Which did you include in your answer? Each term in the sequence of moves is 1 less than the corresponding term in the sequence of differences. You can find any term in the sequence of moves using the formula an = 2n - 1

In the previous task you saw that the number of moves to solve the Tower of Hanoi with n disks given by the equation an = 2n - 1 What is the fewest number of moves required for 7 disks?

127

1. First, consider the number of moves as a sequence: 1, 3, 7, 15, 31, 63, ... To find a pattern, look at the differences between consecutive terms. These differences also form a sequence. 3 - 1 = 2, 7 - 3 = 4, ... Complete the sequence: 2. This sequence contains numbers that are powers of

1. 8, 16, 32 2. 2

The Tower of Hanoi is traditionally seen with three pegs. How would adding more pegs affect the minimum number of moves required to solve for n disks? Why is this the case?

What to include: As the number of pegs added increases, the number of moves required decreases. The extra pegs allow for more places to move disks, reducing the number of intermediate stacking steps. An example would be three disks with four pegs. This requires 5 moves to solve, instead of 7.