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hw7 - g n 2 = 2 g n 1 6 g n ∀ n ≥ 5 Find an explicit...

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CSE 21: Problem Set 7 Note: This homework is due on May 31 st in class. Unlike those on WeBWorK, you need to submit your solutions in paper , and you also need to show your work to get full credit. 1. Prove by induction that n X k =1 k 3 = ± n + 1 2 ² 2 . 2. F n , n 0 , are the Fibonacci numbers: F 0 = 0 , F 1 = 1 , F n +2 = F n +1 + F n , n 0 . Prove by induction that F 2 n - F n +1 F n - 1 = ( - 1) n +1 , n 1 . 3. Find an explicit solution for the recurrence { g ( n ) } n 0 satisfying g (0) = 1 , g ( n + 1) = 3 g ( n ) - n, n 0 . 4. Find an explicit solution for the recurrence { g ( n ) } n 0 satisfying g (0) = - 1 , g (1) = 3
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Unformatted text preview: , g ( n + 2) = 2 g ( n + 1) + 6 g ( n ) , ∀ n ≥ . 5. Find an explicit solution for the recurrence { h ( n ) } n ≥ satisfying h (0) = 1 , h (1) = 4 , h ( n + 2) = 4 h ( n + 1)-4 h ( n ) , ∀ n ≥ . * ( extra credit ) F n , n ≥ , are the Fibonacci numbers: F = 0 , F 1 = 1 , F n +2 = F n +1 + F n , ∀ n ≥ . Prove that n X k =0 ± n + k 2 k ² = F 2 n +1 , ∀ n ≥ . 1...
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