Starting with the vertices [math]P_1=\left(0,1\right),P_2=\left(1,1\right),P_3=\left(1,0\right),P_4=\left(0,0\right)[/math] of a square, we construct further points as follows:[br][list][*][math]P_5[/math] is the midpoint of [math]P_1P_2[/math],[/*][*][math]P_6[/math] is the midpoint of [math]P_2P_3[/math],[/*][*][math]P_7[/math] is the midpoint of [math]P_3P_4[/math], and so on. [/*][/list]The polygonal spiral path [math]P_1P_2P_3P_4...[/math] approaches a point [math]P[/math] inside the square, as shown in the animation below. You can pause the animation and activate the box [b]Value[/b] to analyse the values of [math]P_n[/math].

[b]1.[/b] If the coordinates of [math]P_n[/math] are [math]\left(x_n,y_n\right)[/math], show that [math]\frac{1}{2}x_n+x_{n+1}+x_{n+2}+x_{n+3}=2[/math] and find a similar equation for the [math]y[/math]-coordinates. [br][br][b]2.[/b] Find the coordinates of [math]P[/math].