q9-induction

# q9-induction - n 2 5 n 1 is odd 1 Moral Basis step could...

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CSE 260 QUIZ-9– Mathematical Induction: ANSWERS (25 minutes) NAME: 1. (10 points) Prove by mathematical induction that p ( n ) : n 2 > n + 1 for n 2 Basis Step: p(2): 2 2 > 2 + 1 4 > 3 is true Induction Step: p ( n + 1) : ( n + 1) 2 > ( n + 1) + 1 Left hand side of the equality: ( n + 1) 2 = n 2 + 2 n + 1 > ( n + 1) + 2 n + 1 by induction hypothesis. ( n + 1) + 2 n + 1 = ( n + 1) + 1 + 2 n > ( n + 1) + 1 n > 2 which is the right hand side of the equality. 2. (10 points) Consider the proposition p ( n ) : n 2 + 5 n + 1 is even (a) Prove that the truth of p(k) implies the truth of p(k+1) when k is positive integer p ( k + 1) : ( k + 1) 2 + 5( k + 1) + 1 Expanding and regrouping, we get k 2 + 2 k + 1 + 5 k + 1 + 1 = ( k 2 + 5 k + 1) + 2 k + 2 is even because ( k 2 + 5 k + 1) is even by the hypothesis and 2 k + 2 is even. (b) For which values of n is p(n) actually true? What is the moral of this exercise. p ( n ) is not true for any values of n. That is, n 2 + 5 n + 1 is odd for all values of n. assume n is even: n 2 is even, 5n is even. therefore, n 2 + 5 n + 1 odd. assume n is odd: n 2 is odd, 5n is odd. therefore,

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Unformatted text preview: n 2 + 5 n + 1 is odd. 1 Moral: Basis step could not be found. That is why p(n) could not be proved. 3. (10 points) Use mathematical induction to prove that 1 . 2 + 2 . 3 + ... + n ( n + 1) = n ( n + 1)( n + 2) / 3 p ( n ) : 1 . 2 + 2 . 3 + ... + n ( n + 1) = n ( n + 1)( n + 2) / 3 Basis Step: p (1) : 1 . 2 = 1(1 + 1)(1 + 2) / 3 Left hand side of the equality= 2 Right hand side of the equality= 1.2.3/3=2 Induction Step: Show p ( k ) → p ( k + 1) for 1 ≤ k ≤ n p ( k +1) : 1 . 2+2 . 3+ ... + k ( k +1)+( k +1)( k +2) = ( k +1)( k +2)( k +3) / 3 Lefthand side of the equality: 1 . 2 + 2 . 3 + ... + k ( k + 1) + ( k + 1)( k + 2) = k ( k + 1)( k + 2) / 3 + ( k + 1)( k + 2) by the hypothesis = k ( k + 1)( k + 2) / 3 + 3( k + 1)( k + 2) / 3 = ( k + 1)( k + 2)( k + 3) / 3 This equal to the right hand side of the equality. 2...
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q9-induction - n 2 5 n 1 is odd 1 Moral Basis step could...

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