working on practices

By reading the textbook, the simple induction and complete induction seem not that confusing. To make sure that i understand the definition and examples, i redid the proof in lecture and in the textbook. It's really important to find the right base case: is this case the very fundamental of the problem? do we need more base cases?
The scratch work helps a lot because it's always convinent to make changes when i look backwards in the problem. All of the mathematical problems we've encountered in this course are solvable. The key is always to know what kind of strategy to use corresponding to the problem and follow it step by step.
For example, we want to find out the possible unit digit that 3^n can have by using simple induction. The method is specified as simple induction. Then we just need to figure out the base case and the induction step. It's easy to find the P(0). Then we have to determine P(n) and let it imply to P(n+1). There are different predicates in different questions. We must make sure we understand how does P(n) work here so that it leads exactly to P(n+1). It's really helpful to correspond our assumption to its inductive case and understand the property. When transforming 3^(n+1) into 3*3^n, slight change is made but huge differnce occurs. We now can see the answers explicitly because 3^n had been extracted. This means we have proved our inductive case correctly.