key5fa00 - COT3100.01, Fall 2000 S. Lang Solution Key to...

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COT3100.01, Fall 2000 S. Lang Solution Key to Assignment #5 (40 pts.) 11/21 Use induction to prove each of the following statements: Proof: We use induction on n 2. (Basis Step) Consider n = 2. LHS. 3 1 12 4 12 5 9 12 5 4 3 ) 1 2 )( 2 ( 2 1 ) 2 ( 2 4 3 RHS . 3 1 1 2 1 1 1 LHS 2 2 2 2 = = = - = - = + + - = = - = - = = i i Thus, the Basis Step is proved. (Induction Hypothesis) Consider n = k . Assume . 2 some for , ) 1 ( 2 1 2 4 3 1 1 2 2 + + - = - = k k k k i k i (Induction Step) Consider n = k + 1. We need to prove (1). - - - ) 2 )( 1 ( 2 3 2 4 3 ) 1 1 )( 1 ( 2 1 ) 1 ( 2 4 3 1 1 1 2 2 + + + - = + + + + + - = - + = k k k k k k i k i (1). of RHS ) 2 )( 1 ( 2 ) 3 2 ( 4 3 ) 2 )( 1 ( 2 ) 3 2 ( 4 3 ) 2 )( 1 ( 2 3 2 4 3 ) 2 )( 1 ( 2 2 2 ) 2 5 2 ( 4 3 r denominato common the using by , ) 2 )( 1 ( 2 ) 1 ( 2 ) 2 )( 1 2 ( 4 3 ) 2 ( 1 ) 1 ( 2 1 2 4 3 2 1 ) 1 ( 2 1 2 4 3 Hypothesis Induction by the , 1 ) 1 ( 1 ) 1 ( 2 1 2 4 3 summation of definition by the , 1 ) 1 ( 1 1 1 (1) of LHS The 2 2 2 2 2 2 2 = + + + - = + + + - = + + + - = + + - - + + - = + + + - + + - = + + + + - = + + + + - = - + + + + - = - + + - = = k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k i k i Thus, the Induction Step is proved. By induction, we proved that the following identity holds for all
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key5fa00 - COT3100.01, Fall 2000 S. Lang Solution Key to...

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