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Discrete MathematicsProve the following by using the principle of mathematical induction for all n \in N: 1^3 + 2^3 + 3^3 + \dots + n^3 = \left(\frac{n(n+1)}{2}\right)^2202314mDesign and Analysis of Algorithms Solve the following recurrence by successive substitution method: f(1) = 1 \quad \text{if } n=1 f(n) = 3f(n/2) + 6 \quad \text{if } n > 120227mDesign & Analysis of AlgorithmsDiscuss the steps in mathematical analysis for recursive algorithm. Do the same for finding the factorial of a number?20197mDesign and Analysis of Algorithms What is the Master's Theorem? State the general form and the three cases of the theorem. Use it to solve the following recurrence relations: (i) T(n) = 2T(n/2) + n (ii) T(n) = 3T(n/2) + n^2 (iii) T(n) = 2T(n/2) + n/\log n Explain which case applies in each situation.20247m
PreviousIf a and b are positive integers, then prove that there exists integers s and t such that \text{gcd}(a, b) = sa + tb.NextWrite short notes on the following: (i) Forward proof (ii) Disjunctive and conjunctive normal form (iii) Fundamental theorem of arithmetic This question covers topics from different modules: * (i) Forward proof: This falls under Module 3: Propositional Logic & Proof Techniques. * (ii) Disjunctive and conjunctive normal form: This also falls under Module 3: Propositional Logic & Proof Techniques, as these are concepts in propositional logic. * (iii) Fundamental theorem of arithmetic: This is a concept from number theory, which is typically covered in Module 2: Mathematical Induction & Counting Techniques (as number theory is often grouped with induction and counting in discrete mathematics courses).
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