hw1-sol - CSC236H: Introduction to the Theory of...

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CSC236H: Introduction to the Theory of Computatoin Homework 1 Solutions 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 . Solution. We prove this by induction on n . Base case: Note that for n = 1 both sides of the equation equal 1 / 2. Inductive hypothesis: Let m N and suppose the statement is true for n = m - 1, i.e. suppose that m - 1 X k =1 1 k ( k + 1) = m - 1 m . Induction step: Using the inductive hypothesis we have m X k =1 1 k ( k + 1) = m - 1 X k =1 1 k ( k + 1) + 1 m ( m + 1) = m - 1 m + 1 m ( m + 1) = ( m - 1)( m + 1) + 1 m ( m + 1) = m m + 1 . Therefore the statement is also true for n = m . 2. Use induction to prove 3 n < n ! for all n > 6 with n N . Solution. Base case: The claim is made for n > 6. Therefore the base case we need to verify is n = 7. Since 3 7 = 2187 and 7! = 5040 the statement holds for the base case. Inductive hypothesis: Let
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hw1-sol - CSC236H: Introduction to the Theory of...

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