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Mastering Mathematical Induction for Proof Verification

Learn the powerful technique of mathematical induction to prove statements in a step-by-step approach. This course covers the fundamentals of induction, inductive base, inductive step, and proof verification. Practice solving various induction problems with detailed explanations and examples. Gain confidence in proving mathematical statements using induction method.

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Mastering Mathematical Induction for Proof Verification

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  1. Section 2.2 Induction Proof by induction:to prove p(n) true, n Idea: like playing the domino game. Suppose dominos are placed correctly, then hitting the 1st domino, when it falls, we know the rest of them will also fall. ( the 2nd will fall, in general, when the kth one falls, it implies the (k+1)th falls. Since k is any arbitrary number, actually every domino falls.)

  2. Induction: Step1: (inductive base) or IB is to show that p(1) true Step2: (inductive step) or IS is to show that p(k)  p(k + 1), where p(k) is called inductive hypothesis or IH.

  3. e.g. show 1+3+5+…(2n-1) = n2, n  1 IB: when n = 1, LHS = 1 = 12 = RHS IS: Assume 1+3+5+….(2k-1)=k2 and want to Show 1+3+5+…(2k-1) + [ 2(k+1)-1] =(k+1)2 1+3+5+…(2k –1) + (2k + 1) = k2 + 2k+ 1 = (k +1)2 k2 (by IH)

  4. n(n+1) 2 e.g. Show 1 + 2 + 3 + … n = ,  n  1 IB: when n = 1, LHS = 1 = = RHS IS: Assume 1 + 2 + 3 + …k = and want to show 1 + 2 + 3 + …+ k + (k+1) = 1(1+1) 2 k(k+1) 2 (k+1)(k+2) 2

  5. k(k+1) 2 1 + 2 + 3 + …+ k + (k+1) = + (k+1) = = by direct proof: Let x = 1 + 2 + 3 + …+ k +) x = k + k-1 + k-2 + … + 1 2x = (k+1)+(k+1)+ … +(k+1)  x = k(k+1) 2 (by IH) k(k+1) + 2(k+1) 2 (k+1) (k+2) 2 k of them k(k+1) 2

  6. eg: Show 1+ 21+ 22 + … + 2n = 2n+1 – 1,  n  IB: when n = 1, LHS = 1+ 2 = 22 - 1 = RHS IS: Assume 1 + 2 + 22 + … 2k = 2k+1 – 1, and Show 1 + 2 + 22 + … 2k + 2k+1 = 2k+2 – 1  1 + 2 + 22 + … 2k + 2k+1 = 2k+1 – 1 + 2k+1 = 2*2k+1 – 1 = 2k+2 – 1 2k+1– 1 (by IH)

  7. eg: Show 22n – 1 is divisible by 3,  n  1 IB: when n = 1, LHS = 22 – 1 = 3 divisible by 3 IS: Assume 22k – 1 is divisible by 3 (22k – 1 = 3m for some integer m), and want to show 22(k+1) – 1 is divisible by 3.  22(k+1) – 1 = 22k+2 – 1 = 22 * 22k – 1 = 22(3m+1) – 1 ( 22k – 1 = 3m 22k = 3m + 1) =12m + 3 = 3(4m + 1) where 4m + 1 is an integer  is divisible by 3

  8. k__ k+1 (by IH)

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