ss_11_4

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 oftheformula1+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. Toprovethat1+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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This note was uploaded on 12/11/2011 for the course MAC 1140 taught by Professor Kutter during the Fall '11 term at FSU.

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ss_11_4 - Section 11.4 Mathematical Induction Mathematical...

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