Problem Set 3

Finding the closed form for the given function was tricky. Unwinding it in the conventional sense didn’t quite lead to the closed form. The crucial steps towards finding the closed form were done after unwinding and recognizing the patter in the un-winded terms. After unwinding and simplifying, I realized that each term was a multiple of 3. More specifically, the ratio between two consecutive term is 3. Thus the expansion was that of a geometric series. Thus to find the closed from of G(n), I had to use the geometric series formula, and plug in the ratio and the first term in the series. After finding the closed form of G(n), proving it using simple induction was pretty straightforward.

First Assignment

The most challenging question of the first assignment would have to be question 4. I didn’t know how to approach the question to begin with. But after wrestling with it for awhile and getting help on it, the question started to make sense to me. Proving the contradiction was a bit tricky, because we had to choose the well ordering principle. I had to prove that there exists a smallest numerator for a quotient representing the golden ratio. This led to the required contradiction proving that the golden ratio is an irrational number, which I used to answer the last part of the question: proving 5^1/2 is irrational.

Second Problem Set

The first part of the problem set where we had to prove that for n > N, n postage can be made with 5-cent and 11-cent stamps was pretty straight forward, since we did a similar question during lecture. Proving the second part of that question was a bit more difficult than I initially thought.

I wrote a small java app to figure out what was the smallest n that the postage can be made consecutively after. The magic number we were looking for was 39. Proving that 39=11x +5y has no solution on the natural number line was a little trouble at first. I was trying to come up with an elaborate proof, but later realized a simple simple proof would suffice. I went to the TA office hour and he mentioned that all that the proof requires are some cases to illustrate that 39=11x +5y has no natural number solutions.

I formulated the proof focusing on 11x, because it has fewer cases than 5y.