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MIT6_042JS10_lec08_prob

# MIT6_042JS10_lec08_prob - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science February 22 Prof. Albert R. Meyer revised February 18, 2010, 16 minutes In-Class Problems Week 4, Mon. Problem 1. Prove by induction: 1 1 1 1 1 + 4 + 9 + · · · + n 2 < 2 n , (1) for all n > 1 . Problem 2. (a) Prove by induction that a 2 n × 2 n courtyard with a 1 × 1 statue of Bill in a corner can be covered with L-shaped tiles. (Do not assume or reprove the (stronger) result of Theorem 6.1.2 that Bill can be placed anywhere. The point of this problem is to show a different induction hypothesis that works.) (b) Use the result of part ( a ) to prove the original claim that there is a tiling with Bill in the middle. Problem 3. n Find the ﬂaw in the following bogus proof that a = 1 for all nonnegative integers n , whenever a is a nonzero real number. Bogus proof. The proof is by induction on n , with hypothesis P ( n ) ::= k n. a k = 1 , where k is a nonnegative integer valued variable.

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MIT6_042JS10_lec08_prob - Massachusetts Institute of...

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