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Prove the following: 1) 1+3+5...+ (2n-1) = n²

Prove the following: 1) 1+3+5...+ (2n-1) = n²

College Algebra
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter8: Sequences And Series
Section8.5: Mathematical Induction
Problem 1E: Mathematical induction is a method of proving that a statement P(n) is true for all ________ numbers...
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Prove the following: 1) 1+3+5... + (2n-1) = n² 2) 2° +2¹+2²+ ... + 2n = 2n+1 − 1 Remember: 2⁰=1 Sometimes a statement is not valid for P(1), P(2), .... But valid at another starting point. Example: when is n! > 2n for every integer n> prove n! > 2n
Transcribed Image Text:Prove the following: 1) 1+3+5... + (2n-1) = n² 2) 2° +2¹+2²+ ... + 2n = 2n+1 − 1 Remember: 2⁰=1 Sometimes a statement is not valid for P(1), P(2), .... But valid at another starting point. Example: when is n! > 2n for every integer n> prove n! > 2n
Mathematical Induction is a proof technique used to prove the statement in the form: Vn E N, P (n) There are two parts to a proof by induction: Basis Step: Show P(n) is true if P(1) is true Inductive Step: Show the statement Vk € N, P(k) ⇒ P(k + 1) is true
Transcribed Image Text:Mathematical Induction is a proof technique used to prove the statement in the form: Vn E N, P (n) There are two parts to a proof by induction: Basis Step: Show P(n) is true if P(1) is true Inductive Step: Show the statement Vk € N, P(k) ⇒ P(k + 1) is true
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