A grasshopper jumps along a line. His first jump takes him 1 cm., the second 2 cm, etc. Each jump can take him to the right or to the left. Prove that after 2017 jumps, the grasshopper cannot be in the same location it started.

The grasshopper cannot be in the same location it started because by 2017, it would have covered a distance of 1017576.5 to either left or right. and that distance is not an integer.

Explanation:

The grasshopper jumps in integer of 1 cm to either right or left

Calculating for the total distance covered in 2017 jumps

Total Distance = (2017 x 2018)/2 = 2035153 cm

To either right or left: this implies that the total distance be divided by 2

=> 1017576.5 cm

It shows that at some point, the grasshopper will not be in a position of integer on either right or left.

Therefore, it can not get to its initial point of start.