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hw1KEY - COP 3503 Practice Problems#1 Induction Proofs...

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COP 3503 – Practice Problems #1 – Induction Proofs ANSWERS 1. = - + = - n 1 i 3 1 ) 1 n 4 )( 1 n ( n ) 1 i 2 ( i 2 base case: n=1 LHS: ( 29 ( 29 [ ] = = = - = - 1 1 i 2 ) 1 ( 2 1 ) 1 ( 2 1 2 1 i 2 i 2 RHS: 2 3 6 ) 3 )( 2 )( 1 ( ) 1 ) 1 ( 4 )( 1 1 )( 1 ( 3 1 3 1 = = = - + LHS = RHS so base case is true inductive hypothesis: assume conjecture is true for n=k. Thus, = - + = - + = - + = - k 1 i 2 3 3 1 2 3 1 3 1 ) k k 3 k 4 ( ) 1 k 3 k 4 ( k ) 1 k 4 )( 1 k ( k ) 1 i 2 ( i 2 inductive step: prove conjecture true for n=k+1. Prove: + = + + + = - + + + + = - 1 k 1 i 3 1 3 1 ) 3 k 4 )( 2 k )( 1 k ( ) 1 ) 1 k ( 4 )( 1 ) 1 k )(( 1 k ( ) 1 i 2 ( i 2 RHS: Expanding gives: ) 6 k 17 k 15 k 4 ( ) 6 k 11 k 4 k 6 k 11 k 4 ( ) 6 k 11 k 4 )( 1 k ( 2 3 3 1 2 2 3 3 1 2 3 1 + + + = + + + + + = + + + LHS: ( 29 ( 29 [ ] + = = + + + - + = - + + + - = - 1 k 1 i k 1 i 2 3 3 1 ) 1 k 2 )( 2 k 2 ( ) k k 3 k 4 ( 1 ) 1 k ( 2 1 k 2 1 i 2 i 2 ) 1 i 2 ( i 2 ) 6 k 18 k 12 ( ) k k 3 4 ( ) 2 k 6 k 4 ( ) k k 3 k 4 ( 2 3 3 2 3 3 1 2 2 3 3 1 + + + - + = + + + - + = 3 6 k 17 k 15 k 4 3 6 k 18 k 12 k k 3 k 4 2 3 2 2 3 + + + = + + + - + = ) 6 k 17 15 k 4 ( 2 3 3 1 + + + = LHS = RHS so the conjecture is proven true
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2. = + + = + n 1 i 3 4 2 n 1 n n 2 i 2 i 2 ) )( ( ) ( base case: n=1 LHS: ( 29 ( 29 [ ] [ ] = = = + = + = + 1 1 i 8 4 2 2 2 2 2 1 2 1 2 2 i 2 i 2 ) ( ) ( RHS: 8 3 24 3 2 1 2 1 1 1 1 3 4 3 4 = = = + + ) )( )( ( ) )( )( ( LHS = RHS so base case is true
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