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ACTIVITY 3: MATHEMATICAL INDUCTION INSTRUCTIONS: Use the Principle of Mathematical Induction to verify the following assumptions, you may use any value to the predicate function to test and prove if n is element of natural numbers. Given 1: Use the Principle of Mathematical Induction to prove that, for n natural number, 6" -1 is divisible by 5. ANSWER: For any n 2 1, let P(n) be the statement that 6n - 1 is divisible by 5. Basis Step: Show that P(1) is true. Inductive Step: Prove that n 2 1, and suppose that P(n) holds, that is, 6n - 1 is divisible by 5. It remains to show that P(n+1) holds, that is, that 6"* - 1 is divisible by 5. Conclusion: End of Proof. Given 2: Use the Principle of Mathematical Induction to prove that 12 +2"+3' + ...+ n' =n (n + 1) (2n +1)/6 ANSWER: For any n 2 1, let P(n) be the statement that 6n - 1 is divisible by 5. Basis Step: Show that P (1) is true. Inductive Step: Assume that P(n) is true and show that P (n + 1) is true by adding (n + 1) 2 to both sides of the equation. Conclusion: End of Proof