Homework Answers
T(n) = 2T(n/2) + n T(n) = 2(2T(n/4) + n/2) + n T(n) = 4T(n/4) + n + n T(n) = 8T(n/8) + n + n + n ..... ..... ..... T(n) = nT(n/n) + n + ......... + n + n + n T(n) = n + n + ......... + n + n + n [log(n) times] T(n) = n log(n) T(n) = O(n log(n))
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2) (3 pts) Use mathematical induction to show that when n is an exact power of...
Use mathematical induction to show that when n is an exact power of 2, the solution...
Use mathematical induction to show that when n is an exact power of 2, the solution of the recurrence: { if n 2 2 T(n) for k> 1 if n 2 T(n) 2T(n/2) is T(n) n log
Use the Principle of Mathematical Induction to prove that (2i+3) = n(n + 4) for all...
Use the Principle of Mathematical Induction to prove that (2i+3) = n(n + 4) for all n > 1.
Prove by mathematical induction (discrete mathematics) n? - 2*n-1 > 0 n> 3
Prove by mathematical induction (discrete mathematics)
n? - 2*n-1 > 0 n> 3
6. Use Mathematical Induction to show that (21 - 1)(2i+1) n for all integers n >...
6. Use Mathematical Induction to show that (21 - 1)(2i+1) n for all integers n > 1. 2n +1 (5 marks) i=1
Use the Principle of mathematical induction to prove 2. Use the Principle of Mathematical Induction to...
Use the Principle of mathematical induction to prove
2. Use the Principle of Mathematical Induction to prove: Lemma. Let n E N with n > 2, and let al, aa-.., an E Z all be nonzero. If gcd(ai ,aj) = 1 for all i fj, then gcd(aia2an-1,an)1. 1, a2,, an
8. Use mathematical induction to prove that n + + 7n 15 3 5 is an...
8. Use mathematical induction to prove that n + + 7n 15 3 5 is an integer for all integers n > 0.
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.
Problem 3. Find the exact solutions to the following recurrences and prove your solutions using induction...
Problem 3. Find the exact solutions to the following recurrences and prove your solutions using induction 1, T(1) = 5 and T(n) T(n-1) + 7 for all n > 1. 2. T (1)-3 and T(n)-2T(n-1).
(a) Use mathematical induction to prove that for all integers n > 6, 3" <n! Show...
(a) Use mathematical induction to prove that for all integers n > 6, 3" <n! Show all your work. (b) Let S be the subset of the set of ordered pairs of integers defined recursively by: Basis Step: (0,0) ES, Recursive Step: If (a, b) ES, then (a +2,5+3) ES and (a +3,+2) ES. Use structural induction to prove that 5 (a + b), whenever (a, b) E S. Show all your work.
(5) Use induction to show that Ig(n) <n for all n > 1.
(5) Use induction to show that Ig(n) <n for all n > 1.
Use mathematical induction to show that when n is an exact power of 2, the solution of the recurrence: { if n 2 2 T(n) for k> 1 if n 2 T(n) 2T(n/2) is T(n) n log
Use the Principle of Mathematical Induction to prove that (2i+3) = n(n + 4) for all n > 1.
Prove by mathematical induction (discrete mathematics)
n? - 2*n-1 > 0 n> 3
6. Use Mathematical Induction to show that (21 - 1)(2i+1) n for all integers n > 1. 2n +1 (5 marks) i=1
Use the Principle of mathematical induction to prove
2. Use the Principle of Mathematical Induction to prove: Lemma. Let n E N with n > 2, and let al, aa-.., an E Z all be nonzero. If gcd(ai ,aj) = 1 for all i fj, then gcd(aia2an-1,an)1. 1, a2,, an
8. Use mathematical induction to prove that n + + 7n 15 3 5 is an integer for all integers n > 0.
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.
Problem 3. Find the exact solutions to the following recurrences and prove your solutions using induction 1, T(1) = 5 and T(n) T(n-1) + 7 for all n > 1. 2. T (1)-3 and T(n)-2T(n-1).
(a) Use mathematical induction to prove that for all integers n > 6, 3" <n! Show all your work. (b) Let S be the subset of the set of ordered pairs of integers defined recursively by: Basis Step: (0,0) ES, Recursive Step: If (a, b) ES, then (a +2,5+3) ES and (a +3,+2) ES. Use structural induction to prove that 5 (a + b), whenever (a, b) E S. Show all your work.
(5) Use induction to show that Ig(n) <n for all n > 1.
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