Proof by Induction

Proof by induction is one of the most powerful methods of proof, allowing an observation of a single instance to be applied to all possible instances.  The relation of inductive proofs to the area of computer science can be seen in their close resemblance to recursion.

A proof by induction always involves three parts.  These are: the basis, the inductive hypothesis, and the inductive step.

• Basis Step: This is a property of a particular number (n) that can be expressed as a function of that number ( P(n) ).
• Inductive Hypothesis:  Here the hypothesis is made that P(n) not only holds for the particular n of the basis step, but for all n in a given range, say n greater or equal to zero.
• Inductive Step: This is where it is shown (usually by a substitution of (n+1) for n in the function P(n) or the addition of a (n+1)th term that the inductive hypothesis is true.

The following is a proof by induction that the summation of the powers of 2 through some number n is equal to 2^(n+1) - 1.