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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 speci±ed 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 ±rst 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 .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/01/2011 for the course YEAR 1 taught by Professor Various during the Three '11 term at University of Sydney.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online