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Proof by Induction

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									Proof by Induction
There are many relationships (formulas) in mathematics which include indices. One of the most
common type is formula associated with sequences and series. For example the formulas for the partial
sums of series.
                                n
 For an arithmetic series Sn  2t1  (n 1)d
                                2
                               t (1 r n )
  For a geometric series Sn  1
                                  1 r
Here n is an index that can take on the value of any positive integer.
                                     
Mathematicians use proof by induction to show that these formulas are true for any positive integer
value for n.
                                    
Proof by induction works on the same principle as the domino chain. You set up a string of dominos on
end close enough together so that each domino hits and knocks down the domino that follows it. Then
you push down only the first domino and all of the others fall down in order. The relationship is if the
kth domino falls then the k+1st domino will fall; each domino pushes the next one. If I push the first, it
pushes the second, and the second pushes the third and the third pushes the fourth etc.
Induction Principle
  1. Show that the relationship you are proving is true for the smallest value of the index
     (If the first domino is pushed it will fall)
  2. Show that if the relationship is true for an arbitrary value of the index(k for examp le) that implies
that it is true for the next value of the index (k+1)
   (If the kth domino falls the k+1st domino will fall)
     Since n=1 is true and this implies n=2 is true then if n=2 is true this implies n=3 is true etc.
  3. The relationship is then true for all values of the index.
     (All the dominos will fall)
Example 1: A law of exponents states that ar an  arn Use an induction proof to show this is true for
all positive integers n.
1. Show it is true for n=1           ar a1  ar1
                  ar is r factors of a multiplied together.     Definition of exponents
                    1                                        
                  a is 1 factor of a                             Definition of exponent=1
                  ara1 is r+1 factors of a multiplied together   Definition multiplication
                                                 
                  r+1 factors of a multiplied together is ar+1    Definition of exponents QED
2. Assume it is true for n=k (any arbitrary value of the index)
               If ar ak  ark is true show this implies ar ak1  ark1
                            Start with ar ak  ark              Assumed true
                            a a a a a
                              r k 1      rk 1
                                                                 Multiplication property of equality
                            
                             a   a a
                               r  k 1
                                 aa         rk 1
                                                                 Associative property multiplication
                                                                     
                             a a a
                               r k1     rk1
                                                               From step one. QED
Example 2: Another law of exponents states ar   arn . Use an induction proof to show that this is
                                                        n


true for all positive integers n.   
                                    

                                                         
Example 3: Prove by induction that the partial sum formula for an arithmetic series is true for all
                                   n
positive integers n, that is Sn  2t1  (n 1)d
                                   2
1. Show for n=1
                      1
                 S1  2t1  (11)d 
                      2
                                      
                      1
                 S1  (2t1 )
                      2
                 S1  t1
         The one term sum is equal to the first term QED
                                   k
2. Assume true for n=k Sk  2t1  (k 1)d and show this implies the formula is true for k+1
                  k 1        2
  That is Sk 1        2t1  (k)d
                    2

               Sk 1  Sk  t k 1   
               t k 1  t1  kd
                      
                        k
                Sk 1  (2t1  (k 1)d)  t1  kd
                        2
                        2t k  k(k 1)d  2t1  2kd
                Sk 1  1
                                         2
                        2t k  2t1  k 2 d  kd  2kd
                Sk 1  1
                                        2
                        (k  1)2t1  (k 2  k)d
                Sk 1 
                                   2
                        k 1
                Sk 1        2t1  kd
                          2
                                            QED.
Example 4: Prove by induction that the partial sum formula for an geometric series is true for all
                                     t1 (1 r n )
positive integers n, that is n 
                               S
                                         1 r



                                     

								
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