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### Quiz4s

Course: COMP 2300, Fall 2009
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2300 COMP Discrete Structures for Computing, Fall 2008 October 27, 2008 Quiz #4 (Solution) Consider the recursively defined sequence a1 = 1, an = an-1 + 2n - 1, n Z, n 2 (a) Compute the first five terms of the sequence (3 pts.) a1 = 1 a2 = a1 + 2 2 - 1 = 1 + 4 - 1 = 4 a3 = a2 + 2 3 - 1 = 4 + 6 - 1 = 9 a4 = a3 + 2 4 - 1 = 9 + 8 - 1 = 16 a5 = a4 + 2 5 - 1 = 16 + 10 - 1 = 25 (b) Discover the explicit formula...

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2300 COMP Discrete Structures for Computing, Fall 2008 October 27, 2008 Quiz #4 (Solution) Consider the recursively defined sequence a1 = 1, an = an-1 + 2n - 1, n Z, n 2 (a) Compute the first five terms of the sequence (3 pts.) a1 = 1 a2 = a1 + 2 2 - 1 = 1 + 4 - 1 = 4 a3 = a2 + 2 3 - 1 = 4 + 6 - 1 = 9 a4 = a3 + 2 4 - 1 = 9 + 8 - 1 = 16 a5 = a4 + 2 5 - 1 = 16 + 10 - 1 = 25 (b) Discover the explicit formula for an , using the iterative method (3 pts.) an = n2 (c) Prove by mathematical induction that formula the discovered in (b) is correct (4 pts.) (Use the back of the sheet if necessary) Base case: n = 1 a1 = 1 = 12 , i.e. the claim is true in the base case. Inductive step: Assume ak = k 2 for some k Z, k 1. Consider ak+1 . According to the recursive formula, ak+1 = ak + 2(k + 1) - 1. But, accor...

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