07 Mathematical Induction

# 07 Mathematical Induction - 7 Mathematical Induction...

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Induction - 1 7 - Mathematical Induction Mathematical induction is one of the most powerful and elegant tools in proving algebraic identities and other logical inferences. The abstract formulation of an inductive process is based on the following fundamental principle: suppose that there exists a sequence of statements, denoted by P(n), n = 1, 2, ... , such that 1. the first statement, P(1 ), is true, and 2. assuming that P(1), P(2), ... , P(k) are all true statements, it can be proved that P(k + 1) is also a true statement. These two conditions imply that every statement P(n) is true. Mathematical induction appears to have been known to the mathematicians of the Hellenistic times. More rigorous accounts of the process were provided several centuries later by F. Maurolico (1494-1575) and B. Pascal (1623-1662). Example Mathematical induction can be used in proving this statement P(n) : For every interger n , 2 ) 1 ( ... 2 1 + = + + + n n n To prove it, first note that P (1) is true. Namely: 2 ) 1 1 ( 1 1 + = Subsequently, assume that the above statement holds for a given interger k . In other words,

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