In the solution to Part 1 we found that Robert would walk miles.
Now let's see how far Derek would walk.
In the first quatrain Derek would walk
Then he would walk 799 miles for half of the additional miles. In other words he would walk miles more, for a total of
Then he would walk 798 miles for a third of the last additional miles (798 times 1/3 of the last term). In other words, miles more, for a total now of
Then adding 797 miles for a fourth of the last additional miles, we get a total of
We can see that eventually, the numerator will decrease to 0 times the previous additional miles ("Until I would walk 0 additional miles"). Factoring out a 800, we get a total distance of
(The next term would be when he reached 0 times the previous additional miles.)
The above can also be written as
Does that look familiar? That's right, it's the Choose function, also known as the binomial coefficient. Then this summation becomes
Notice that the sum of the numbers on any row of Pascal's triangle (values along row for , starting at ) is a power of two:
This can be proven by the binomial expansion of :
Therefore, how far Derek would walk can be written more succinctly.
So Derek would walk miles.
Who gets the girl?
Therefore, Derek gets the girl and would walk miles more than Robert would walk.