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# 15 - LECTURE 15 INDUCTION STRENGTHENED(6.4 I think I should...

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LECTURE 15: INDUCTION STRENGTHENED ( § 6.4) I think I should have no other mortal wants, if I could always have plenty of music. It seems to infuse strength into my limbs and ideas into my brain. Life seems to go on without effort, when I am filled with music. George Eliot (1819 - 1880) 1. Strong Induction Recall our infinite staircase. To climb it, we needed to do two things. First, climb the bottom step. Second, use the fact that we climbed the k th step to climb the ( k + 1)st step. But we are not using everything available to us. We didn’t only climb the k th step, we also climbed all the previous steps. So we can strengthen our inductive case. We do this by not only assuming P ( k ); but rather we assume P ( i ) for all i in the range 1 i k . We still show that we can then get P ( k +1). Why don’t we always do this strong induction? Because sometimes it just complicates things. Example 1. Let A 1 = 1 and A 2 = 3. Then let A n = 2 A n - 1 - A n - 2 when n 3. What sequence do we get? We have A 3 = 2 A 2 - A 1 = 2 · 3 - 1 = 5. Also A 4 = 2 A 3 - A 2

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15 - LECTURE 15 INDUCTION STRENGTHENED(6.4 I think I should...

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