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ss_11_4 - Section 11.4 Mathematical Induction Mathematical...

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Section 11.4 Mathematical Induction Mathematical induction is a method that is useful for proving mathematical formulas that might otherwise be difficult (or impossible) to prove. Example. Show that 1 + 2 + 3 + · · · + n = n ( n +1) 2 for all n . We note the following: n = 1: 1 = 1(2) 2 = 1 n = 2: 1 + 2 = 2(3) 2 = 3 n = 3: 1 + 2 + 3 = 3(4) 2 = 6 · · · We see that it is impossible to directly check all (infinitely many) integers n to prove the validity of the formula 1 + 2 + 3 + · · · + n = n ( n +1) 2 for all n . Thus, some method of proof, other than a direct check, will be needed to establish the formulas’ validity for all n . One method of proof is Mathematical Induction. To prove that 1+2+3+ · · · + n = n ( n +1) 2 by Mathematical Induction, we show that the formula holds when n = 1, and then we show that whenever the formula holds for some integer n , then the formula also holds for the next larger integer n + 1. These two facts imply that the formula holds for all n . The details of the proof will be shown below.
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