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__.