06 Week 6 - CMPUT 272(Stewart Lecture 10 Reading Epp...

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CMPUT 272 (Stewart) Lecture 10 Reading: Epp Chapter 5.1-5.3 Mathematical Induction an additional rule of inference for proving facts about all integers greater than or equal to a certain value When do we use mathematical induction? Consider the following theorem for example. Theorem. For all n Z + , 1 + 2 + · · · + n = n ( n +1) 2 . How to prove it? It’s easy to check for the first few numbers: n = 1: 1 = 1(1+1) 2 n = 2: 1 + 2 = 2(2+1) 2 n = 3: 1 + 2 + 3 = 3(3+1) 2 . . . But we can’t check all positive integers! To prove the theorem, we check that the equality holds for n = 1, and prove that IF it holds for an an arbitrary positive integer THEN it holds for the next integer. 1
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To prove IF the equality holds for an an arbitrary positive integer THEN it holds for the next integer: Let k Z + . Suppose the equality holds for n = k : 1 + 2 + · · · + k = k ( k + 1) 2 . We want to prove that the equality holds for n = k + 1, that is: 1 + 2 + · · · + ( k + 1) = ( k + 1)( k + 2) 2 . Strategy: Write the LHS of what we want to prove in terms of the LHS of the supposition: 1 + 2 + · · · + ( k + 1) = 1 + 2 + · · · + k +( k + 1) = k ( k + 1) 2 +( k + 1) by the equality for n = k = ( k + 1) k 2 + 1 = ( k + 1)( k + 2) 2 the equality holds for n = k + 1 We have proven: for all k Z + : if 1 + 2 + · · · + k = k ( k +1) 2 then 1 + 2 + · · · + ( k + 1) = ( k +1)( k +2) 2 .
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