# Feedback on hw 4

## 3.44

Let S be the set of natural numbers that can be written as a sum of 3s and 10s. Most of you proved that all numbers above 18 were in this set, but some of you forgot to talk about the other numbers below 18. Some wrote down a list of them without any explanation. Please do not forget to justify all your claims !

You not only have to justify why each number is in the set, but also why there aren't any other elements.

In the list of numbers below 18 in S, a few of you forgot to include 13 (this is one reason everything should be proven: avoiding mistakes !). Others included 0, but even though it can be written as an "empty" sum of 3s and 10s, it is not a natural number.

The induction arguments were sometimes more complicated than necessary, e.g. proving three cases at each step. The use of strong induction avoids this (you still have to consider three base cases).

## 3.56

The most common mistake here was in proving the base cases: because the relation on an is only valid when n ≥ 3, two base cases must be considered, one for n = 1 and one for n = 2. Considering the case n = 3 separately is not necessary (but does not hurt).

## 12.19

Same comment as above, two base cases must be considered: one for n = 0, one for n = 1.

## 4.2, 4.13

Most of you got those right. In 4.13, depending on your method of proof you should state somewhere that you are assuming (e.g. by symmetry) that x is positive.