In a line of falling dominoes, each domino has the same experience as the one before it. It is hit, it falls over, and it hits another. In an infinite line of dominoes, only the first one has a unique experience. If we can prove that the first domino will fall, and then prove that any domino will cause the one following it to fall, we can conclude that tipping the first domino will start an endless sequence of falling dominoes.
These, in fact, are the three steps of mathematical induction!
If we suspect that a particular condition is true for an endless sequence of steps, for example, we simply:
1. prove that it is true for some starting step, then
2. prove that, if it is true for any subsequent step, then it must be true for the following step, and then
3. conclude that it is true for all steps from the first one.
As you can see, we count in the same way. We choose a place to start and then keep adding one. With mathematical induction, the steps do not have to be one unit apart.
It is important to understand that mathematical induction is not the same as inductive reasoning. Inductive reasoning is not a proof of anything. It is simply drawing/inferring conclusions from a collection of data. Mathematical Induction, however, is a deductive proof (despite its name). It is also closely related to recursion, which is the basis for fractals. Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs.
How I wish you could be my professor – I have been having a consistent problem with the chain rule for what seems like weeks. I couldn’t grasp it at all. The first couple minutes of your video – it all instantly clicked for me. You are a life saver and a wonderful teacher. Thank you so much for your explanations!!!!!!!
Taikrys (on a CCM YouTube video about the Chain Rule)
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