I have a two quick questions on mathematical induction. I'm new to it, so my answers seem a little shaky. I wanted to see what some of the you guys thought.

Question 1

We're asked to prove using mathematical induction that for all natural numbers ,

Proof (?):

Let

1. Base case: :

CHECK

2. Assume ,

.

Then,

3. Thus, by P.M.I. blah blah QED.

Is this legit? It just seems strange to me since I'm working completely on the right side. Yet any other way I try to attempt seems to lead me down some crazy algebra path (I could very well be doing it wrong).

Question 2

UsegeneralizedPMI principles of mathematical induction to prove

Proof(?):

First, we know that since we're taking a factorial that n must be an integer (this was not originally given to us, so I guess it's implied?).

Let

(I guess that since k is greater than 4, I might rotate that Z pi radians and define k as a natural number?)

1. Base case: k = 4

Good.

2. Mathematical induction:

Assume that for some integer that

Note,

Also note,

And

(since k is always positive.)

So,

3. by general PMI, i'm done?

Again, this feels a bit shaky to me. Perhaps because I dealt with so many pieces individually before putting it all back together.

So again, my question: Am I using mathematical induction properly here? Do my proofs prove what they are supposed to?