L8 - Induction: Review When using Mathematical Induction to...

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Induction: Review When using Mathematical Induction to prove that a property P(n) holds for all positive integers n, we must do the following two steps: 1)Basis step: Show that P(1) is true. 2)Demonstrate that if P(k) is true for arbitrary k, then P(k+1) is also true. (This is done by assuming that P(k) is true, and using it to prove that P(k+1) is also true.)
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Sum of Evens Prove that the sum of the first n even numbers is n 2 +n. 2 11 Proof: First we can write the sum of the first n positve even numbers as 2 . So P(n): 2 nn ii i i n n    1 2 1 Basis: P(1): 2 2 1 1 i i 2 1 Inductive: Assume P(k) is true, i.e. 2 4 ... 2 2 .(*) k i k i k k   1 2 1 We will show P(k+1) is true, i.e. 2 4 ... 2 2 2 2 ( 1) ( 1). k i k k i k k     2 Adding 2k+2 to both sides of (*), 2 4 . .. 2 2 2 2 2. k k k k k       2 ( 2 1) ( k k k 2 ( ( kk We have now shown P(k+1) is true so the proof is complete.
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Example Prove that 1+2+2
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L8 - Induction: Review When using Mathematical Induction to...

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