# 10 - The University of Sydney MATH 1004 Second Semester...

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The University of Sydney MATH 1004 Second Semester Discrete Mathematics 2011 Tutorial 10 Week 11 Use induction to prove the following propositions. For each of the given propositions, we let S ( n ) be the given proposition, which is to be proved true for all integers greater than or equal to some specified integer, n 0 say. Then we show that (a) S ( n 0 ) is true. (b) ( n n 0 ) ( S ( n ) S ( n + 1) ) is true. [To show (b), we suppose that S ( n ) is true and, assuming that n n 0 , we prove that S ( n + 1) is true.] Then we conclude that S ( n ) is true for all positive integers n n 0 . 1. Prove that 2 n n + 12, for all integers n 4. 2. Prove that 1 + 3 + 5 + · · · + (2 n - 1) = n 2 , for all positive integers n . 3. Prove that the sum of the first n positive even integers is n 2 + n . 4. Prove that 2 + 5 + 8 + · · · + (3 n - 1) = n (3 n + 1) 2 , for all positive integers n . 5. Prove that 6 divides n ( n 2 + 5) for all positive integers n . 6. Prove that 11 n - 4 n is divisible by 7 for all positive integers n . 7. Prove that 5 n - 4 n - 1 is divisible by 16 for all positive integers n 8. Prove that for any integer n 1, (2 n )! 2 n is an integer.

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