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11.4 Mathematical Induction

Explore the concept of mathematical induction and learn how to prove statements for all positive integers. Follow the step-by-step process with examples.

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11.4 Mathematical Induction

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  1. 11.4 Mathematical Induction Objective: To prove a statement is true for all positive integers.

  2. What is mathematical induction? • It is taking a statement and proving it is true for all positive integers.

  3. Mathematical Induction • If we are given a series (summation of an infinite sequence), we want to be able to prove that the statement is true for all values of n. • Ex 1: Sn = 1 + 2 + 3 + … + n = n(n+1) 2 …What are the steps to the proof?

  4. Proof by Mathematical Induction • Let Snbe a statement involving positive integer n. • Step 1: Show that S1 is true. • substitute n=1 and show it is true • Step 2: Assume Sk is true. • write the statement “Assume (substitute n=k) is true.” • Step 3: Show Sk+1 is true. • start with statement from step 2 and add the next term (the “k+1” term) • Replace the first part with the assumption from step 2 • Manipulate the LEFT SIDE ONLY to become the same as the right side

  5. Back to Ex1. Use mathematical induction to prove that 1 + 2 + 3 + …+ n = n(n+1)/2 for all n. • Step 1: Show S1 is true. • Step 2: Assume Sk • Step 3: Prove Sk+1

  6. Ex2. Use mathematical induction to prove: • Step 1: Show S1 is true. • Step 2: Assume Sk • Step 3: Prove Sk+1

  7. Ex3. Use mathematical induction to prove: • Step 1: Show S1 is true. • Step 2: Assume Sk • Step 3: Prove Sk+1

  8. You try. Use mathematical induction to prove: • Step 1: Show S1 is true. • Step 2: Assume Sk • Step 3: Prove Sk+1

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