On a coordinate plane lets name a move dot A. The dot A moves each day. On day 1, it moves 1 in the x- axis direction. On day 2 it moves $2^2$ in the y- axis direction. On day 3 it moves -$3^2$ in the x- axis direction. On day 4 it moves -$4^2$ in the y axis direction. And so on therefore on day n it moves ±$n^2$ spaces in either x or y axis direction.
Here is the question. How would you simplify the coordinates of dot A on Day n. For the coordinates you must use N and it must be a formula that can work for any natural number n may be also, it must be generalized not something that would be long ex) the x coordinate could be $1^2$ -$3^2$+$5^2$....
Please solve this and if you please, write how you got to the answer. Thank you in advance.
Hint Since the terms have alternating sign you essentially want a simple form for expressions of the form $\sum_{n=0}^m ( (4n+1)^2 - (4n+3)^2 )$ and since $(4n+1)^2 - (4n+3)^2 = -4(4n+1)-4$ you only need to know $\sum_{n=0}^m n$ to reduce your sum to a simple form.