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M135F07A2

# M135F07A2 - MATH 135 Assignment#2 Fall 2007 Due Wednesday...

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MATH 135 Fall 2007 Assignment #2 Due: Wednesday 26 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. A sequence of integers is defined by x 1 = 2, x 2 = 86, x m +2 = - 2 x m +1 + 15 x m for m 1. Prove that x n = 4 · 3 n + 2( - 5) n for all n P . 2. A sequence of integers is defined by y 1 = 2, y 2 = 11, y 3 = 2, y m +3 = 3 y m +1 - 2 y m for m 1. Prove that y n = ( - 2) n + 3 n + 1 for all n P . 3. Suppose that x, y R . Prove that x 2 - 4 xy + 5 y 2 - 4 y + 4 = 0 if and only if ( x, y ) = (4 , 2). 4. Prove by contradiction that log 10 3 cannot be written in the form m n where m and n are positive integers. 5. Prove that if x = y and a = 0, then x y = x + a y + a . 6. To prove a statement of the form “If H then C 1 or C 2 ” (where H, C 1 , C 2 are mathematical statements), we can assume that H is TRUE and that C 1 is FALSE, and prove that C 2 is TRUE. (If C 1 happened to be TRUE, we would be done, so we assume that C 1 is FALSE and prove that C 2 must be TRUE.) Suppose that m and n are integers.

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M135F07A2 - MATH 135 Assignment#2 Fall 2007 Due Wednesday...

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