do the following moves in a coordinate system: start at the origin facing in the positive y direction and move 1 unit. For each subsequent move, it is to turn 90 degrees clockwise and go forward 1 unit farther than it went in the previous move. What is the sum of the coordinates of the robot's position after it moved 14 units in one direction?

take some graph paper and follow the moves:

make a list of
move# -- ordered pair --- sum of coordinates
until you see a pattern

1 -- (0,1) -- 1 ...... up 1 -- add 1 to y
2 -- (2,1) -- 3 ...... right 2 -- add 2 to x
3 -- (2,-3) -- -1 ... down 4 -- add -4 to y
4 -- (-6,-3) -- -9 ... left 8 -- add -8 to x
5 -- (-6,13) -- 7 ... up 16 -- add 16 to y
6 -- (26,13) -- 39 ..right 32 -- add 32 to x
...
..

what do you think?

To determine the coordinates of the robot's position after it moves 14 units in one direction, we need to determine the x and y coordinates at each step.

Let's break down the steps:

Step 1: Start at the origin (0, 0) facing in the positive y direction and move 1 unit. The new position is (0, 1).

Step 2: Turn 90 degrees clockwise and go forward 1 unit farther than the previous move. The new position is (-1, 1).

Step 3: Turn 90 degrees clockwise and go forward 1 unit farther than the previous move. The new position is (-1, 2).

Step 4: Turn 90 degrees clockwise and go forward 1 unit farther than the previous move. The new position is (0, 2).

Step 5: Turn 90 degrees clockwise and go forward 1 unit farther than the previous move. The new position is (0, 3).

Step 6: Turn 90 degrees clockwise and go forward 1 unit farther than the previous move. The new position is (-1, 3).

Step 7: Turn 90 degrees clockwise and go forward 1 unit farther than the previous move. The new position is (-1, 4).

Step 8: Turn 90 degrees clockwise and go forward 1 unit farther than the previous move. The new position is (0, 4).

Step 9: Turn 90 degrees clockwise and go forward 1 unit farther than the previous move. The new position is (0, 5).

Step 10: Turn 90 degrees clockwise and go forward 1 unit farther than the previous move. The new position is (-1, 5).

Step 11: Turn 90 degrees clockwise and go forward 1 unit farther than the previous move. The new position is (-1, 6).

Step 12: Turn 90 degrees clockwise and go forward 1 unit farther than the previous move. The new position is (0, 6).

Step 13: Turn 90 degrees clockwise and go forward 1 unit farther than the previous move. The new position is (0, 7).

Step 14: Turn 90 degrees clockwise and go forward 1 unit farther than the previous move. The new position is (-1, 7).

Therefore, the sum of the coordinates of the robot's position after moving 14 units in one direction is (-1 + 0) + (7 + 0) = -1 + 7 = 6.

To determine the sum of the coordinates of the robot's position after it moved 14 units in one direction, we can break down the problem into smaller steps:

1. Start at the origin (0, 0) facing in the positive y direction.
2. Move 1 unit forward to coordinate (0, 1).
3. Turn 90 degrees clockwise.
4. Go forward 1 unit more than the previous move, which would be 2 units.
5. Move 2 units forward to coordinate (-2, 3).
6. Turn 90 degrees clockwise.
7. Go forward 1 unit more than the previous move, which would be 3 units.
8. Move 3 units forward to coordinate (1, 6).

Now, let's observe a pattern in the robot's movements:
- After the 1st move, the robot's coordinates are (0, 1).
- After the 2nd move, the robot's coordinates are (-2, 3).
- After the 3rd move, the robot's coordinates are (1, 6).

By analyzing the pattern, we can see that the robot's x-coordinate changes in the order of 0, -2, 1, -3, 2, -4, 3, -5, 4, -6, 5, -7, 6, -8. While the y-coordinate follows a pattern of 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105.

Since the robot is moving 14 units in one direction, we can find the coordinates corresponding to the 14th movement.

To find the x-coordinate of the 14th move, we can calculate:
x-coordinate = 14 / 2 * (-1)^[14 / 2]
= 14 / 2 * (-1)^[7]
= 7 * (-1)
= -7

To find the y-coordinate of the 14th move, we can calculate:
y-coordinate = 14 * (14 + 1) / 2
= 14 * 15 / 2
= 210 / 2
= 105

Therefore, after moving 14 units in one direction, the robot's coordinates are (-7, 105).

The sum of the coordinates would be:
-7 + 105 = 98.