M135F07A1 - MATH 135 Fall 2007 Assignment #1 Due: Wednesday...

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Unformatted text preview: MATH 135 Fall 2007 Assignment #1 Due: Wednesday 19 September 2007, 8:20 a.m. N.B. Assignments 3 to 9 will not be distributed in class. You must download them from the course Web site. Hand-In Problems 1. Disprove the statement “There is no positive integer n > 3 such that n 2 + ( n + 1) 2 is a perfect square”. 2. Prove that 0 + 4 + 11 + ··· + 3 n 2- n- 2 2 = n ( n- 1)( n + 2) 2 for every positive integer n . 3. Prove that if x 6 = 1, then x + 3 x 2 + 5 x 3 + ··· + (2 n- 1) x n = x + x 2- (2 n + 1) x n +1 + (2 n- 1) x n +2 (1- x ) 2 for every positive integer n . 4. Using the fact that d dx ( x ) = 1 and the Product Rule, prove by induction that d dx ( x n ) = nx n- 1 for every positive integer n . 5. (a) Prove algebraically that 1 n- 1 n + 1 ≥ 1 ( n + 1) 2 for all positive integers n . (b) Prove by mathematical induction that 1 + 1 2 2 + 1 3 2 + ··· + 1 n 2 ≤ 2- 1 n for all positive integers n ....
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This note was uploaded on 10/24/2009 for the course MATH 135 taught by Professor Andrewchilds during the Winter '08 term at Waterloo.

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M135F07A1 - MATH 135 Fall 2007 Assignment #1 Due: Wednesday...

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