M325K_HW08_Soln - last to next writing k k k k LHS = = = =...

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M325K HW 08 - Solutions to Graded Questions (total= 22pts+3free=25 pts) Section 3.8 15. (3pts) gcd(832, 10933) =13: because using Euclidean Algorithm: 10933= 832 (13)+117 gcd(10933, 832) =gcd(832, 117) 832= 117(7)+13 =gcd(117, 13) 117=13(9)+0 =gcd(13,0)=13 Section 4.1 26. (2pts) 11 4 3 4 ) 3 1 ( ) 3 0 ( ] 3 ) 1 [( 3 2 2 2 1 1 2 = + + = + + + + + = + = k k 41. (2pts) = + 1 0 )! 1 ( n k k k n . (Note: There are more than one correct answer.) 56. (3pts) Given = 1 1 2 ) ( n i i n i . Transform by making the change of variable . 1 = i j When 0 , 1 = = j i and when 2 , 1 = = n j n i . Also since 1 + = j i , given sum= = + 2 0 2 ) 1 ( 1 n j j n j Section 4.2 12. (6pts) Proof (by mathematical induction): Let the property P( n ) be the equation 1 ) 1 ( 1 3 2 1 2 1 1 + = + + + + n n n n " Base case: Prove P( n ) for n =1: LHS= 2 1 1 and RHS= 1 1 1 + . They are equal. Inductive hypothesis: For any integer 1 k assume P( k ) is true. That is, assume 1 ) 1 ( 1 3 2 1 2 1 1 + = + + + + k k k k " Inductive step: Prove the property for 1 + = k n . We need to prove the following equation: 2 1 ) 2 )( 1 ( 1 3 2 1 2 1 1 + + = + + + + + k k k k " .
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RHS k k k k k k k k k k k k k k k k hypotesis inductive by k k k k separately term
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Unformatted text preview: last to next writing k k k k LHS = + + = + + + + = + + + + = + + + + + + = + + + + = + + + + ⋅ + + ⋅ + ⋅ = 2 1 ) 2 )( 1 ( ) 1 )( 1 ( ) 2 )( 1 ( 1 2 ) 2 )( 1 ( 1 ) 2 )( 1 ( ) 2 ( ) 2 )( 1 ( 1 1 ) 2 )( 1 ( 1 ) 1 ( 1 3 2 1 2 1 1 2 " Section 4.3 18. (6pts) Proof (by mathematical induction): Let the property P( n ) be “ n n 6 9 5 < + ” Base case: Prove P( n ) for n =2: LHS= 9 5 2 + and RHS= 2 6 . Thus 36 6 34 9 5 2 2 = < = + is true. Inductive hypothesis: For any integer 2 ≥ k assume P( k ) is true. That is, assume k k 6 9 5 < + Inductive step: Prove the property for 1 + = k n . That is we need to prove 1 1 6 9 5 + + < + k k . . RHS LHS k k k k k k = = ⋅ < ⋅ < + = + < + = + + + 1 1 1 6 6 6 6 5 ) 9 5 ( 5 45 5 9 5 (The first inequality is because 45 9 < and 2nd inequality is by inductive hypothesis. )...
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This note was uploaded on 04/29/2010 for the course MATH m325k taught by Professor Seckin during the Spring '10 term at University of Texas-Tyler.

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M325K_HW08_Soln - last to next writing k k k k LHS = = = =...

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