Vikram the mathematical ant is scurrying across a very large piece of graph paper, which has x and y axes marked on it. Vikram starts at the origin (0, 0) and moves as follows:

stage 1: he moves 1 unit in the direction of the positive y axis

stage 2: turns 90° to his left and moves 2 units in that direction

stage 3: turns 90° to his left and moves 3 units in that direction

stage 4: turns 90° to his left and moves 4 units in that direction

Vikram now continues this sequence of moves: at each stage he turns 90° to his left and moves a distance which is one unit more than the step before.

a        What will Vikram’s co-ordinates be after 202 stages?

It takes Vikram one second to move one unit on the graph paper.

b       How long does it take Vikram to reach the point (60, –60)?

c        Can you give a general method for working out what Vikram’s co-ordinates will be after N stages?