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Mathematical Induction • A common proof technique • Let P be a proposition to prove, and express P in terms of a positive integer parameter n. To show that P(n) is true, follow these steps: 1. Base Step: Verify that P(1) is true 2. Induction Hypothesis: Assume that P(n) is true for some n 1 3. Inductive Step: Show that P(n + 1) is also Mathematical Induction • Note that the inductive step is the only step that really requires real work. In particular, – The base step usually only requires simple checking on what happens when n = 1 – No proof or computation is required for the induction hypothesis. We simply proclaim the assumption that P(n) is true – In the inductive step, we want to prove that P(n + 1) is true. Typically, this is done by applying the information gathered from P(n). In other words, we are trying to show that P(n) P(n + 1) Mathematical Induction – Example • Prove by mathematical induction that n f(n) = i = n (n + 1) 2 i=1 where n > 0 Example • Prove by mathematical induction that the sum of the first n positive odd numbers is n2. • Hint: note that the nth odd number is simply 2n - 1. So, all you have to prove is the following: