Answered

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Renae has been playing Tower of Hanoi and has noticed that the minimum number of moves it takes to defeat the game is related to the number of disks she must move. She has recorded her observations below. Write an equation that describes this pattern. Show your work.


Number of Disks in the Tower ║Minimum Number of Moves

1 ║1

2║ 3

3 ║7

4 ║15

5 ║31

Sagot :

MrRoyal

Answer:

[tex]y = 2^x - 1[/tex]

Step-by-step explanation:

Let the number of disks be x and the minimum number of moves be y.

[tex]y = 2^x - 1[/tex]

When x = 1: y = 1

[tex]y = 1 = 2 - 1 = 2^1 - 1[/tex]

When x = 2: y = 3

[tex]y = 3 = 4 - 1 = 2^2 - 1[/tex]

When x = 3: y = 7

[tex]y = 7 = 8 - 1 = 2^3 - 1[/tex]

When x=4: y = 15

[tex]y = 15 = 16 - 1 = 2^4 - 1[/tex]

When x = 5: y = 31

[tex]y = 31 = 32 - 1 = 2^5 - 1[/tex]

Notice that for each derived expression of y, the exponent of 2 is being replaced by the value of x.

The general equation is: [tex]y = 2^x - 1[/tex]

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