hw1 - cos x y = cos x cos y-sin x sin y sin x y = sin x cos...

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CSC236H: Introduction to the Theory of Computatoin Homework 1 Due on Tuesday January 26, 2010 1. Use induction to prove that the following equation holds for all positive integers n : n X k =1 1 k ( k + 1) = n n + 1 . 2. Use induction to prove 3 n < n ! for all n > 6 with n N . 3. Use induction to prove that every positive integer n Z + can be expressed as the product of an odd number and a power of two. In other words, show n = (2 k + 1)2 p where k N and p N . Hint: In the inductive step, break the proof down into one case for even numbers and one case for odd numbers. 4. If you’ve done much graphics programming, you may have used the following two standard trigono- metric identities often used to rotate objects:
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Unformatted text preview: cos( x + y ) = cos( x ) cos( y )-sin( x ) sin( y ) sin( x + y ) = sin( x ) cos( y ) + cos( x ) sin( y ) Using these formulas, prove by induction that for any natural number n and any real number x (cos( x ) + i sin( x )) n = cos( nx ) + i sin( nx ) where i is the imaginary unit , i.e. the square root of -1 (in other words i 2 =-1). Do not convert this problem to exponential notation and try to solve it in that form. It doesn’t make things simpler and it will confuse us when we grade. Hint: this is not as bad as it looks. If your equations seem very complicated and won’t simplify, look for a mistake in your algebra. 1...
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