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The Connection between Sequences and Terms

4.14.3 The Connection between Sequences and Terms

4.14.3 • The Connection between Sequences and Terms

Activity

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Some checkers are lined up, with blue on one side, red on the other, and 1 empty space between them. A move in this checker game pushes any checker forward 1 space, or jumps over any 1 checker of the other color. Jumping the same color is not allowed, moving backward is not allowed, and 2 checkers cannot occupy the same space.

A row of six circles: the first three circles are blue, a gap, then the last three circles are red. All circles are outlined in black and set against a green background.

You complete the puzzle by switching the colors completely: ending up with black on the right, red on the left, and 1 empty space between them.

1. Using 1 checker on each side, complete the puzzle. What is the smallest number of moves needed?

Compare your answer:

3 moves

2. Using 3 checkers on each side, complete the puzzle. What is the smallest number of moves needed?

Compare your answer:

15 moves

3. Estimate the number of moves needed if there are 2 or 4 checkers on each side. Then test your guesses.

Compare your answer:

8 moves and 24 moves

4. Noah says he used the solution for 3 checkers on each side to help him solve the puzzle for 4 checkers. Describe how this might happen.

Compare your answer:

The moves that don’t involve the fourth checker on each side are the same as before, so it is possible to decide the number of moves by running the first half of the three-checker game, then moving the fourth checkers, and then running the second half.

5. How many moves do you think it will take to complete a puzzle with 7 checkers on each side?

Compare your answer:

63 moves

Self Check

The table below shows the number of checkers and the number of moves needed to win a skipping game.

Number of Checkers 1 2 3 4
Number of Skips Needed 2 8 18 32

How many skips are needed with 6 checkers on each side?

  1. 48
  2. 128
  3. 72
  4. 50

Additional Resources

Using a Table to Find Missing Values

Number of Checkers 1 2 3 4 7 8
Number of Moves Needed 3 8 15 24 63 ?
This table presents the relationship between the number of checkers and the required moves, including a missing value.

What is the pattern that you used in the previous activity to find the number of moves depending on the number of checkers?

Each row is multiplied by 1 more, so:

1·3=31·3=3

2·4=82·4=8

3·5=153·5=15

4·6=244·6=24

How many moves are needed with 5 checkers?

5·7=355·7=35 moves

What about 6 checkers?

6·8=486·8=48 moves

Try it

Using a Table to Find Missing Values

Using the table and the pattern, how many moves are needed with 8 checkers?

Here is how to solve this problem using a general strategy:

The next column in the table had 7 checkers and then multiplied 7 by 9 to get 63 moves.

Next, the 8 would need to be multiplied by 10, so:

8·10=808·10=80

80 moves are needed.