11X1 T14 10 mathematical induction 3 (2011)

Download 11X1 T14 10 mathematical induction 3 (2011)

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<ul><li> 1. Mathematical Induction</li></ul> <p> 2. Mathematical Inductione.g.v Prove 2n n 2 for n 4 3. Mathematical Inductione.g.v Prove 2n n 2 for n 4Step 1: Prove the result is true for n = 5 4. Mathematical Inductione.g.v Prove 2n n 2 for n 4Step 1: Prove the result is true for n = 5LHS 25 32 5. Mathematical Inductione.g.v Prove 2n n 2 for n 4Step 1: Prove the result is true for n = 5LHS 25 RHS 52 32 25 6. Mathematical Inductione.g.v Prove 2n n 2 for n 4Step 1: Prove the result is true for n = 5LHS 25 RHS 52 32 25 LHS RHS 7. Mathematical Inductione.g.v Prove 2n n 2 for n 4Step 1: Prove the result is true for n = 5LHS 25 RHS 52 32 25 LHS RHS Hence the result is true for n = 5 8. Mathematical Inductione.g.v Prove 2n n 2 for n 4Step 1: Prove the result is true for n = 5LHS 25 RHS 52 32 25 LHS RHS Hence the result is true for n = 5Step 2: Assume the result is true for n = k, where k is a positiveinteger &gt; 4i.e. 2k k 2 9. Mathematical Inductione.g.v Prove 2n n 2 for n 4Step 1: Prove the result is true for n = 5LHS 25RHS 52 32 25 LHS RHS Hence the result is true for n = 5Step 2: Assume the result is true for n = k, where k is a positiveinteger &gt; 4i.e. 2k k 2Step 3: Prove the result is true for n = k + 1 k 1 k 12i.e. Prove : 2 10. Proof: 11. Proof: 2 k 1 12. Proof: 2 k 1 2 2k 13. Proof: 2 k 1 2 2k 2k 2 14. Proof: 2 k 1 2 2k 2k 2 k2 k2 15. Proof: 2 k 1 2 2k 2k 2 k2 k2 k2 k k 16. Proof: 2 k 1 2 2k 2k 2 k2 k2 k2 k k k 2 4k 17. Proof: 2 k 1 2 2k 2k 2 k2 k2 k2 k k k 2 4k k 4 18. Proof: 2 k 1 2 2k 2k 2 k2 k2 k2 k k k 2 4k k 4 k 2 2k 2k 19. Proof: 2 k 1 2 2k 2k 2 k2 k2 k2 k k k 2 4k k 4 k 2 2k 2k k 2 2k 8 20. Proof: 2 k 1 2 2k 2k 2 k2 k2 k2 k k k 2 4k k 4 k 2 2k 2k k 2 2k 8 k 4 21. Proof: 2 k 1 2 2k 2k 2 k2 k2 k2 k k k 2 4k k 4 k 2 2k 2k k 2 2k 8 k 4 k 2 2k 1 22. Proof: 2 k 1 2 2k 2k 2 k2 k2 k2 k k k 2 4k k 4 k 2 2k 2k k 2 2k 8 k 4 k 2 2k 1 k 12 23. Proof: 2 k 1 2 2k 2k 2 k2 k2 k2 k k k 2 4k k 4 k 2 2k 2k k 2 2k 8 k 4 k 2 2k 1 k 12 2 k 1 k 1 2 24. Proof:2 k 1 2 2k 2k 2 k2 k2 k2 k k k 2 4k k 4 k 2 2k 2k k 2 2k 8 k 4 k 2 2k 1 k 1 2 2 k 1k 1 2Hence the result is true for n = k + 1 if it is also true for n = k 25. Proof: 2 k 1 2 2k 2k 2 k2 k2 k2 k k k 2 4k k 4 k 2 2k 2k k 2 2k 8 k 4 k 2 2k 1 k 12 2 k 1 k 1 2 Hence the result is true for n = k + 1 if it is also true for n = kStep 4: Since the result is true for n = 5, then the result is true forall positive integral values of n &gt; 4 by induction . 26. Proof: 2 k 1 2 2k 2k 2 k2 k2 k2 k k k 2 4k k 4 Exercise 6N; k 2 2k 2k6 abc, 8a, 15 k 2 2k 8 k 4 k 2 2k 1 k 12 2 k 1 k 1 2 Hence the result is true for n = k + 1 if it is also true for n = kStep 4: Since the result is true for n = 5, then the result is true forall positive integral values of n &gt; 4 by induction .</p>