Section 6.4, due on February 14th

Section 6.4 The Strong Principle of Mathematical Induction!

What was difficult for me?
The Strong Principle of Mathematical Induction - for each positive integer n, let P(n) be a statement.  If

1. P(1) is true and
2. the implication "If P(i) for every integer i with 1 < i < k, then P(k+1)" is true for every positive integer k, 

then P(n) is true for every positive integer n.

In a recursively defined sequence {An}, only the first term or perhaps the first few terms are defined specifically, say a1, a2, ...., ak for some fixed k \in \textbf{N}.

I don't really understand the definition of a recursive sequence.  The definition in the book is difficult to understand.  I look forward to learning more about this in class tomorrow and understanding how it works.

At first, I didn't understand what the difference between the Strong Principle of Mathematical Induction.  Then I looked online, and went to a website math.stackexchange.com and I found this:

"The only difference between regular induction and strong induction is that in strong induction you assume that every number up to k satisfies the condition that you wish to prove whereas in regular induction you only assume that some integer k satisfies this condition."

Based on that explanation, that is what I understand the Strong Principle to be.  It picks an m and a k, and proves that if m is true, then all values leading up to k are true.  Interesting stuff.

What did I find interesting?
You know, at first thought I really didn't think there was going to be too much difference between the Principle of Mathematical Induction and the Strong Principle of Mathematical Induction.  But after some analysis and thinking through things, I realize that there indeed is a difference.  With normal PMI, you prove a point is true and then prove that the point directly after it is true as well.  With Strong PMI, you prove that a point is true, and then prove that ALL points leading up to another point k are true.

I can see how that would be helpful.  Instead of proving the Domino method with just two dominoes, you can prove that if there is a domino is one spot, and then that there is another domino in another spot, you can prove that all the dominos between your two original dominos will fall.  Now that is cool stuff.