On a coordinate grid, Paul draws a line segment of length 1 from the origin to the right, stopping at $(1,0)$. He then draws a line segment of length 2 up from this point, stopping at $(1,2)$. He continues to draw line segments to the right and up, increasing the length of the line segment he draws by 1 each time. One of his line segments stops at the point $(529,506)$. What is the endpoint of the next line segment that he draws?

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-042.jpg?height=312&width=404&top_left_y=2056&top_left_x=1316)
(A) $(529,552)$
(B) $(576,506)$
(C) $(575,506)$
(D) $(529,576)$
(E) $(576,552)$

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-043.jpg?height=193&width=195&top_left_y=234&top_left_x=390)
We begin by listing the first few line segment endpoints as determined by the number of segments that Paul has drawn.

| Line Segment Number | Endpoint of the Line Segment |
| :---: | :---: |
| 1 | $(1,0)$ |
| 2 | $(1,2)$ |
| 3 | $(4,2)$ |
| 4 | $(4,6)$ |
| 5 | $(9,6)$ |
| 6 | $(9,12)$ |
| 8 | $(16,12)$ |

Since each new line segment is drawn either vertically or horizontally from the endpoint of the previous line segment, then only one of the $x$-coordinate or $y$-coordinate changes from one endpoint to the next endpoint.

Further, since the odd numbered line segments are horizontal, the $x$-coordinate of the endpoint of these segments changes from the previous endpoint while the $y$-coordinate remains the same.

Similarly, since the even numbered line segments are vertical, the $y$-coordinate of the endpoint of these segments changes from the previous endpoint while the $x$-coordinate remains the same.

We are given that one of the line segments ends at the point $(529,506)$.

We will begin by determining the number of segments that must be drawn for the $x$-coordinate of the endpoint of the final line segment to be 529 .

The first line segment is drawn horizontally from the origin, to the right, has length 1 , and thus ends with an $x$-coordinate of 1 .

The third line segment is drawn horizontally to the right, has length 3 and thus ends with an $x$-coordinate of $1+3=4$.

The fifth line segment is drawn horizontally to the right, has length 5 and thus ends with an $x$-coordinate of $1+3+5=9$.

To get an idea of how many line segments are required before the $x$-coordinate of the endpoint is 529 , we begin by considering the first 21 line segments.

The endpoint of the 21 st line segment has $x$-coordinate equal to $1+3+5+\cdots+17+19+21$. To make this sum easier to determine, we rearrange the terms to get $(1+21)+(3+19)+(5+17)+(7+15)+(9+13)+11=22+22+22+22+22+11=22 \times 5+11=121$. Since 121 is much smaller than 529 , we continue to try larger numbers of line segments until we finally reach 45 line segments.

The endpoint of the 45 th line segment has $x$-coordinate equal to $1+3+5+\cdots+41+43+45$. To make this sum easier to determine, we rearrange the terms to get

$$
(1+45)+(3+43)+(5+41)+\cdots+(19+27)+(21+25)+23=46 \times 11+23=529
$$

Therefore, the 45th line segment ends with an $x$-coordinate of 529 .

The next line segment (the 46th) will also have an endpoint with this same $x$-coordinate, 529 , since even numbered line segments are vertical (thus, only the $y$-coordinate will change).

This tells us that $(529,506)$ is the endpoint of either the 45th or the 46th line segment.

At this point we may confirm that 506 is the $y$-coordinate of the endpoint of the 45th line segment (and hence the 44th line segment as well).

The second line segment is drawn vertically from the $x$-axis, upward, has length 2 , and thus ends with $y$-coordinate 2 .

The fourth line segment is drawn vertically upward, has length 4 and thus ends with a $y$-coordinate of $2+4=6$.

The sixth line segment is drawn vertically upward, has length 6 and thus ends with a $y$-coordinate of $2+4+6=12$.

The endpoint of the 45 th line segment has $y$-coordinate equal to $2+4+6+\cdots+40+42+44$. Rearranging the terms of this sum, we get

$$
(2+44)+(4+42)+(6+40)+\cdots+(20+26)+(22+24)=46 \times 11=506
$$

Thus, $(529,506)$ is the endpoint of the 45 th line segment.

The 46th line segment is vertical and has length 46 .

Therefore the $y$-coordinate of the endpoint of the 46th line segment is

$$
2+4+6+\cdots+40+42+44+46=506+46=552
$$

Since the $x$-coordinate of the endpoint of the 46th line segment is the same as that of the 45th, then the endpoint of the next line segment that Paul draws is $(529,552)$.