Unformatted text preview: n + 1) (by the induction hypothesis P ( n )) = n ( n + 1) + 2( n + 1) 2 = ( n + 2)( n + 1) 2 = ( n + 1)(( n + 1) + 1) 2 . Then P ( n + 1) follows by the transitivity of equality. Since we proved this by assuming P ( n ), we have P ( n ) = ⇒ P ( n + 1). Since n was arbitrary, we have ∀ n ( P ( n ) = ⇒ P ( n + 1)) Since we proved P (0) and ∀ n ( P ( n ) = ⇒ P ( n + 1)), the statement ∀ nP ( n ) follows by the principle of Mathematical Induction, and the theorem follows. 1...
View
Full Document
 Fall '11
 Mitchelle
 Mathematical Induction, Recursion, Inductive Reasoning, Mathematical logic, Mathematical proof, Structural induction

Click to edit the document details