key4fa02 - COT3100C-01, Fall 2002 S. Lang Solution Keys to...

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COT3100C-01, Fall 2002 S. Lang Solution Keys to Assignment #4 (40 pts.) 10/24/2002 Use induction to prove each of the following questions, and be sure to mark clearly when and where the induction hypothesis is applied in each question: 1. (10 pts.) . 16 5 ) 1 4 ( 5 5 , 1 integer for that Prove 1 1 + - = + = n j n n n j j Answer : We use induction on n 1. (Basis Step) Consider n = 1. In this case, the LHS = . 5 5 1 5 1 1 1 = = = j j j The RHS = . 5 16 80 16 5 3 25 16 5 ) 1 4 ( 5 1 1 = = + = + - + Thus, LHS = RHS, so the Basis Step is proved (Induction Hypothesis) Consider n = k . Suppose 16 5 ) 1 4 ( 5 5 1 1 + - = + = k j k k j j for some k 1. (Induction Step) Consider n = k +1. We need to prove (1) - - - 16 5 ) 3 4 ( 5 16 5 ) 1 ) 1 ( 4 ( 5 5 2 1 1 1 1 + + = + - + = + + + + = k k j k k k j j (1). of RHS 16 5 ) 3 4 ( 5 16 5 ) 3 4 ( 5 5 16 5 ) 15 20 ( 5 16 5 ) 16 16 1 4 ( 5 16 5 ) 1 ( 16 5 ) 1 4 ( 5 Hypothesis Induction by the , 5 ) 1 ( 16 5 ) 1 4 ( 5 summation of definition by the , 5 ) 1 ( 5 (1) of LHS that the Note 2 1 1 1 k 1 1 k 1 1 k 1 1 = + + = + + = + + = + + + - = + + + - = + + + -
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key4fa02 - COT3100C-01, Fall 2002 S. Lang Solution Keys to...

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