Stuck with proof homework!
So I've been looking at these two problems for the last bit and have no idea on how to even approach them. Any help or guidance would be extremely appreciated.
- Prove that if , then:
for every positive integer .
- We are playing a mound-splitting game - we start with a single mound of rocks. In each move, we pick a mound, split it into 2 mounds of arbitrary size (say, and rocks), multiply the # of rocks in the 2 mounds and write down the result ( ), and continue playing until only one rock remains in each mound. At the end, we add up all the numbers written down after the splits. Prove that if we start with rocks, then the final sum will be , no matter how we split the mounds or in which order we split them.