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.
Our daughter Angelina was home schooled and from the start did not like maths. Over the years I have battled through, trying 5 different programs along the way. Angelina was surviving but not enjoying the journey but, when it came to algebra, the future looked dim. A friend recommended Graeme to me as her three sons had been tutored by him. She could not recommend him more highly. My daughter has just finished year 12 maths and did very well, thanks to Graeme (that would be an understatement). Graeme has a love for his subject and a genuine interest in his students. Graeme seems to meets his students where they are and tailors the lessons to meet their individual needs and interests. Graeme not only explains concepts clearly, but we all found him an interesting, knowledgeable and humble man. We are extremely happy that Graeme was recommended to us. Now our daughter is looking forward to uni with a grateful heart. We also could not recommend Graeme more highly.
Angela K (parent, 2013)
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