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Mathematic Induction

There are several different methods for proving things in math.  One type you've probably already seen is the "two column" proofs  you did in Geometry.

In the Algebra world, mathematical induction is the first one you usually learn because it's just a set list of steps you work through.  This makes it easier than the other methods.  There's only one semi-obnoxious step (the main one!)  But, I've got a great way to work through it that makes it a LOT easier.  I was going to start out by officially stating "The Principle of Mathematical Induction"...  But, writing it out on my rough draft even gave ME a headache!  So, I'm just going to write out the steps...  Go ahead and read them through...  But, don't expect to understand anything yet.  I'm going to explain how the whole thing works after.  It really isn't that bad.

The four steps of math induction:

1 )

 Show P( 1 ) is true

Let n = 1 and work it out.

2 )

 Assume P( k ) is true

Stick a k in for all the n's and say it's true.

3 )

 Show P( k ) --> P( k + 1 )
* In math, the arrow  -->  means "implies" or "leads to."

USE P( k ) to show that P( k + 1 ) is true. ... P( k ) is Very important!

4 )

 End the proof

Write "Thus, P( n ) is true." []
This is the modern way to end a proof.

The third step is the only tricky part...  And it's the most important step...  You have to show EVERY little detail!  Remember that you are proving something -- which means that you have to spell out your entire argument.  I call this a "monkey proof."  You have to write it out soooo clearly that the average intelligent monkey can read it through and not get confused.  Teachers are very hip to the fact that omitting details or skipping steps on these is, probably, clueless fudging on your part.

OK, I promised that I would actually explain this thing to you, didn't I?  Take a minute and go back to read the steps again.  I'll wait.

Don't worry.  I didn't understand it at first either.  Lucky for you,  I'm going to explain it so you WILL get it!