Mathematical Induction

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Presentation transcript:

Mathematical Induction PreCalculus 8-4

Mathematical Induction

Mathematical Induction Is this true for all possible sums of odd numbers? Mathematical Induction

Mathematical Induction Start by making a conjecture But, we still don’t know this to be true for ALL possible numbers Mathematical Induction

Mathematical Induction Start by making a conjecture There are an INFINITE number of possibilities to check. We need a process to prove this for ALL numbers Mathematical Induction

Mathematical Induction Mathematical Induction - Method of proving a given property true for a set of numbers by proving it true for 1 and then true for an arbitrary positive integer by assuming the property true for all previous positive integers Induction Step – A logical statement that leads from one true statement to the next Mathematical Induction

Mathematical Induction Instead of proving an infinite number of statements to be true, we prove that if one is true, the next one must be true also Mathematical Induction

Mathematical Induction

Mathematical Induction

Mathematical Induction

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Mathematical Induction Practice – Use induction to show 2 + 6 + 10 + ... + (4n-2) = 2n2 Mathematical Induction

Mathematical Induction Practice – Use induction to show 1 + 2 + 3 + ... + n = ½ n(n + 1) Mathematical Induction

Mathematical Induction Practice – Use induction to show 3 + 6 + 9 + ... + 3n = ½ 3n(n + 1) Mathematical Induction

Mathematical Induction Practice – Use induction to show 5 + 7 + 9 + 11 + ... + (3 + 2n) = n(n + 4) Mathematical Induction

Mathematical Induction

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Mathematical Induction Your book also has formulas for the fourth and fifth power Mathematical Induction

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Mathematical Induction As fate would have it, I could tile the board regardless of where the missing square is. In fact I can do it for an 8 x 8 chessboard, a 16 x 16 chessboard, etc. As long as there is one square missing. In fact, I can even prove that I can tile any 2n x 2n chessboard, with one square missing with my friend angular triomino. Go ahead, and you try to prove it, work in groups Mathematical Induction

Mathematical Induction Base Case: Trivial Inductive Step: Divide the 2n+1 x 2n+1 checkerboard into four equal pieces as shown, then temporarily remove three squares Mathematical Induction

Mathematical Induction As we have four 2n x 2n boards with one square missing in each, we can tile what is left by induction, and then we can tile the removed squares with on angular triomino Mathematical Induction

Mathematical Induction Homework Page 592 7-15odd, 19, 23, 25 Mathematical Induction