Proof By Induction

The difference between Inductive Reasoning and Deductive Reasoning:
Inductive - reason from special case (facts) to generalization (rules).
Deductive - reason from generalization (rules) to special case (facts).

An example of inductive reasoning:
While at the Hall of Fame Game in the Handley Bowl, I noticed in the football program that the inductees in 2001 were from the class of 1971. I noticed that the inductees in 2000 were from the class of 1970 and that the inductees in 1999 were from the class of 1969.
I made the inductive hypothesis that a student athlete must wait 30 years before he or she is inducted into the Handley Hall of Fame.

In 1575, Francesco Maurolico, an Italian mathematician, performed the first formal proof using the principle of mathematical induction.

    Steps involved in a Proof by Induction:
  1. Show that your hypothesis works for n = 1.
  2. Suppose (or assume) that it works for some specific case n = k.
  3. Then show that your hypothesis works for n = k + 1.

Example:

Now prove the following by induction:

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