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MATH135 - Assignment1 Solutions

MATH135 - Assignment1 Solutions - MATH 135 Assignment#1...

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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 = 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 . 6. For each positive integer n , define R n to be the number of regions into which n lines, no two of which are parallel and no three of which intersect at the same point, divide the plane. (a) Determine R 1 , R 2 , R 3 , R 4 , R 5 . (b) Determine a formula for R n and prove by induction that it is true for all positive integers n . 7. A square array of dots with 10 rows and 10 columns is given. Each dot is coloured either blue or red. Whenever two dots of the same colour are adjacent in the same row or column, they are joined by a line segment of the same colour as the dots. If they are adjacent but of different colours, they are then joined by a green line segment. In total, there are 52 red dots. There are 2 red dots at corners, with an additional 16 red dots on the edges of the array. The remainder of the red dots are inside the array. There are 98 green line segments. (a) How many line segments are there in total, including all three possible colours? (b) Each line segment has 2 ends. Thus, the 98 green line segments account for 196 ends: 98 “red ends” (that is, ends at red dots) and 98 “blue ends” (that is, ends at blue dots). Using your knowledge about the distribution of the dots, determine the total number of “red ends” among all of the line segments. (c) How many blue line segments are there? (This problem is not directly related to the course material, but is included to keep your problem solving skills sharp.)
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Recommended Problems 1. Disprove the statement “5 n + n + 1 is divisible by 7 for every positive integer n ”. 2. Find an expression for 1! + 2(2!) + 3(3!) + · · · + n ( n !) and prove that it is correct. 3. Text, page 104, #16 4. Text, page 104, #17 5. Text, page 105, #31 6. Text, page 105, #33 7. Text, page 109, #74
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MATH 135 Fall 2007 Assignment #1 Solutions Hand-In Problems 1. Counterexample If n = 20, then n 2 + ( n + 1) 2 = 20 2 + 21 2 = 400 + 441 = 841 which is a perfect square, so the given statement is FALSE. How did we find this? We could find this by trying positive integer values starting at n = 4 until we obtained a value for n 2 + ( n + 1) 2 that was a perfect square. Alternatively, we might remember that (20 , 21 , 29) forms a Pythagorean triple, because 20 2 + 21 2 = 29 2 .
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