A particle is located on the coordinate plane at $(5,0)$. Each move for the particle consists of a counter-clockwise rotation of $45^{\circ}$ about the origin followed by a translation of $10$ units in the positive x-direction. If the position of the particle after $150$ moves is $(p,q)$, find $|p|+|q|$
I was discussing this problem with one of my classmate, and he said that, let the initial position $z_0= 5$ then the position after n movements would be $$z_n=[z_{n-1} × e^{\frac{i\pi}{4}}]+10$$ My question is, is this correct? If it is correct, then how did he derive it?