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What Is a Sequence?

4.14.2 What Is a Sequence?

4.14.2 • What Is a Sequence?

Activity

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In the Tower of Hanoi puzzle, a set of discs sits on a peg, while there are 2 other empty pegs.

A move in the Tower of Hanoi puzzle involves taking a disc and moving it to another peg. There are two rules:

  • Only move 1 disc at a time.
  • Never put a larger disc on top of a smaller one.

You complete the puzzle by building the complete tower on a different peg from the starting peg.

1. What is the smallest number of moves in which you are able to complete the puzzle with 3 discs?

Compare your answer:

7 moves

2. What is the smallest number of moves in which you are able to complete the puzzle with 4 discs?

Compare your answer:

15 moves

3. Jada says she used the solution for 3 discs to help her solve the puzzle for 4 discs. Describe how this might happen.

Compare your answer:

Jada moved the tower of discs 1 – 3, then moved disc 4, and then moved the remaining discs 1 – 3 onto it.

4. How many moves do you think it will take to complete a puzzle with 5 discs? Explain.

Compare your answer:

31 moves. For example: Double the previous answer, and then add 1. The number of moves is 31 because 2·15+1=312·15+1=31.

5. How many moves do you think it will take to complete a puzzle with 7 discs?

Compare your answer:

127 moves

In this activity, you have started to find a sequence of terms.

Write these definitions in your math notebook.

  • A sequence is a list of numbers, possibly going on forever, such as all the odd positive integers arranged in order: 1, 3, 5, 7, . . . .
  • The term (of a sequence) is one of the numbers in a sequence.

Self Check

The table below shows the number of chips in a game and the number of moves that can be made.

Number of chips 1 2 3 4
Number of possible moves 3 5 7 9

How many moves would be possible with 7 chips?

  1. 13
  2. 15
  3. 14
  4. 11

Additional Resources

Using a Table to Find the Pattern

Sometimes it is helpful to keep a table to find the pattern needed to answer a question.

For the Tower of Hanoi game, enter the number of discs used and the smallest number of moves to solve the puzzle in a table like the one below. Use the answers from the activity.

Number of Discs 3 4 5 6 7 8
Number of Moves to Solve Puzzle 7 15 31 ? 127 ?
Number of moves required to solve a puzzle, likely Tower of Hanoi, based on the number of discs from 3 to 8.

The pattern was to double the previous answer and then add 1.

Using this pattern, complete the table for 6 discs.

2(31)+1=632(31)+1=63

Then use this rule to find how many moves are needed to complete the puzzle with 8 discs.

There were 127 moves for 7 discs.

So, double 127 and add 1: 2(127)+1=2552(127)+1=255.

Try it

Using a Table to Find the Pattern

Now it’s your turn. For the same game, how many moves would it take to solve the puzzle with 9 discs?

Your answers may vary, but here are some examples:

The pattern is still the same. Double the number of moves before and add 1.

With 8 discs, 255 moves were needed.

Double 255, and then add 1:

2(255)+1=5112(255)+1=511