Induction Notes
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1) Induction is the author's favorite proof technique. It involves assuming a statement is true for a base case, and then using that to show the statement is true for all following cases. 2) The basic steps of a proof by induction are: state the problem, state the plan, prove the base case, state the inductive hypothesis, prove the next case, and state what was proved. 3) The example proves that the sum of the first n natural numbers equals n(n+1)/2 by mathematical induction. It shows the base case is true, assumes the inductive hypothesis, and proves the next case to complete the proof.
Induction Notes
- 1. Induction Tyler Murphy March 17, 2014 Induction is my favorite style of proof. For new students, they hate it. But stop for a few minutes, take some deep breaths and think about what’s going on. Most induction proofs (especially those in this class) are very direct and only involve some algebra. First, let’s look at the basic method of giving a proof by induction. 1. State the problem 2. State how you plan to proceed with the proof. 3. State your base case (n = 1) and show that it is true. 4. State your Inductive Hypothesis. (Assume true for k) 5. Prove that the k+1 case works. 6. State what you proved Let me give an example of how induction works. example Prove that for all n ∈ N, n i=1 i = n(n + 1) 2 Realize first what this is. n i=1 i = 1 + 2 + 3 + 4 + · · · + n. So we want to prove 1 + 2 + 3 + 4 + · · · + n = n(n + 1) 2 . 1
- 2. Proof. By Induction. Base Case: n = 1. 1 ? = 1(1 + 1) 2 = 1(2) 2 = 2 2 = 1. Since 1=1, our base case holds. Note that I used ? = instead of = because I was trying to show that the statements on the left and right were equal. Using the = sign right away assumes that they are, which is assuming what you want to prove. Inductive Hypothesis (I.H.): Assume that 1 + 2 + 3 + 4 + · · · + k = k(k + 1) 2 , for k ∈ N. WTS: 1 + 2 + 3 + 4 + · · · + k + (k + 1) = (k + 1)(k + 1 + 1) 2 . Now we have set up the proof. It is time to begin. 1 + 2 + 3 + 4 + · · · + k + (k + 1) = k(k + 1) 2 + (k + 1)(by I.H.) = k(k + 1) 2 + 2(k + 1) 2 = k(k + 1) + 2(k + 1) 2 = (k + 1)(k + 2) 2 = (k + 1)(k + 1 + 1) 2 So we have that 1 + 2 + 3 + 4 + · · · + k + (k + 1) = (k + 1)(k + 1 + 1) 2 , Which is what we wanted to prove. So, we are done. 2