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Say there is a moving dot on the coordinate plane. It starts on the coordinates of (0,0). On the 1st day it moves to (1,0) the next, (1,4) then (-8,4) then, (-8,-12), then, (17,-12) and so on. Now I have figured out that on day N, it moves $n^2$ spaces towards a paricular direction.

This is what I am trying to get. How do I generalize this into a some sort of formuala. So how can you write the coordinates of the moving dot on day N, using the letter N in a simplified way?

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  • $\begingroup$ Typeset your question with MathX $\endgroup$ – Nebo Alex Apr 14 '16 at 10:46
  • $\begingroup$ Done! Typed my question with mathX $\endgroup$ – user331408 Apr 14 '16 at 11:11
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you can see that the distance is increasing by unity firstly from $(0,0)$ to $(0,1)$ it is 1 unit then from $(1,0)$ to $(1,4)$ it is $2$ on next iteration it will be$ 3 ,4 ,5$ and so on it will eventually be $n$ on the Nth day .

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Looking at the pattern, the coordinates are $(x_n,y_n)$ where $x_{2n-1}=x_{2n}=1^2-3^2+5^2-\dots+(-1)^{n+1}(2n-1)^2$ and $y_{2n}=y_{2n+1}=2^2-4^2+\dots+(-1)^{n+1}(2n)^2$. These are easily summed to get $x_{2n-1}=x_{2n}=\frac{1}{2}(-1+(-1)^n)-2(-1)^nn^2$ and $y_{2n}=y_{2n+1}=(-1)^{n+1}2n(n+1)$.

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