VIEWS: 2 PAGES: 2 POSTED ON: 10/4/2010
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 arn Use an induction proof to show this is true for all positive integers n. 1. Show it is true for n=1 ar a1 ar1 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 ark is true show this implies ar ak1 ark1 Start with ar ak ark Assumed true a a a a a r k 1 rk 1 Multiplication property of equality a a a r k 1 aa rk 1 Associative property multiplication a a a r k1 rk1 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 (11)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