Disc-a8

# Disc-a8 - b k The recursion tells us that b k = 11 b k&...

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1. Let the sequence a 0 ; a 1 ; a 2 ; : : : a 0 = 0 and a n = a n 1 + n for n > 0 . Show that a n = n ( n + 1) = 2 for all positive integers n . We show that the formula is true by induction on n . The base case is n = 0 . We have a 0 = 0 and 0 (0 + 1) = 2 = 0 , so the formula is true for n = 0 . In the induction step , we assume that the formula is true for n = k 1 , that is, a k 1 = ( k 1) k= 2 . We want to conclude that it is true for k , so we compute a k = a k 1 + k = ( k 1) k 2 + k = ( k 1) k + 2 k 2 = k 2 + k 2 = k ( k + 1) 2 which is the formula for n = k . 2. Let the sequence b 0 ; b 1 ; b 2 ; : : : b 0 = 1 and b 1 = 1 and the recursion b n = 11 b n 1 10 b n 2 for n > 1 . of this sequence. Guess a formula for b n and prove that formula is correct by mathematical induction . b 0 = 1 b 1 = 1 b 2 = 11 b 1 10 b 0 = 11 10 = 1 b 3 = 11 b 2 10 b 1 = 11 10 = 1 b 4 = 11 b 3 10 b 2 = 11 10 = 1 So it looks like the formula is b n = 1 . The claim is that b n 1 = 1 and b n = 1 for all n ± 1 . The claim is true for n = 1 , the base case. Suppose it is true for n = k 1 . Then b k 2 = 1 and b k 1 = 1 To show it is true for n = k , we must show that b k 1 = 1 and b k = 1 We already know that b k 1 = 1 , so we concentrate

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Unformatted text preview: b k . The recursion tells us that b k = 11 b k & 1 & 10 b k & 2 = 11 & 10 = 1 which is what we wanted to show. 3. Repeat Problem 2 with the same recursion but with the initial values b = 1 and b 1 = 10 . b = 1 b 1 = 10 b 2 = 11 b 1 & 10 b = 110 & 10 = 100 b 3 = 11 b 2 & 10 b 1 = 1100 & 100 = 1000 b 4 = 11 b 3 & 10 b 2 = 11000 & 1000 = 10000 So it looks like the formula is b n = 10 n . Because 10 = 1 and 10 1 = 1 , the formula holds for the base case. For the induction step, we assume that b k & 2 = 10 k & 2 and b k & 1 = 10 k & 1 and compute b k from the recursion: b k = 11 b k & 1 & 10 b k & 2 = 11 ± 10 k & 1 & 10 ± 10 k & 2 = 11 ± 10 k & 1 & 10 k & 1 = 10 ± 10 k & 1 = 10 k which is what we wanted to show....
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Disc-a8 - b k The recursion tells us that b k = 11 b k&...

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