Mathematical Induction and Binomial Theorem: Proofs and Applications, Study notes of Algebra

Download Mathematical Induction and Binomial Theorem: Proofs and Applications and more Study notes Algebra in PDF only on Docsity! 285 L26 Mathematical Induction; Binomial Theorem Mathematical Induction Mathematical induction is a method for proving mathematical statements which involve natural numbers. We denote a statement ( )A n , n is natural. Example: ( )A n : 2 4 6 ... (2 ) ( 1)n n n+ + + + = + , 1, 2, 3,n = … . The Principle of Mathematical Induction: Suppose that the following two conditions are satisfied with regard to a statement ( )A n : 1. ( )A n is true for 0n n= . 2. If ( )A n is true for n k= ( 0k n≥ ), it is also true for 1n k= + . Then ( )A n is true for all natural numbers 0n n≥ . 286 Example: Use mathematical induction to prove that the formulas are valid. ( )A n : ( 1) 1 2 3 ... 2 n n n + + + + + = , 1n ≥ 287 ( )A n : 3 3 3 3 21 2 3 ... (1 2 3 ... )n n+ + + + = + + + + , 1n ≥ 288 We define ! 1 2 ...n n= ⋅ ⋅ ⋅ ( 1)!n + = Example: Use mathematical induction to prove that the statement is true for all natural numbers 3n ≥ . ( )A n : 1! 2nn −>