COT3100Hmk04Sol - COT 3100 Fall 2002 Homework #4 Solutions...

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Unformatted text preview: COT 3100 Fall 2002 Homework #4 Solutions 1) Use induction on n 1 to prove the following summation formula: . 9 ) 2 )( 1 3 ( 2 ) 2 ( 1 + =- + +- =- n n i i n i Base case: n=1 LHS = 2 ) 2 ( 1 0(-2) ) 2 ( 1 1- =- + =- = i i i RHS = 2 9 18 9 4(4) 2 9 ) 2 )( 1 ) 1 ( 3 ( 2 1 1- =- = +- =- + +- + Thus, the formula is true for n=1. Inductive hypothesis: Assume for an arbitrary n=k that . 9 ) 2 )( 1 3 ( 2 ) 2 ( 1 + =- + +- =- k k i i k i Inductive Step: Under that assumption prove for n=k+1 that 9 ) 2 )( 4 3 ( 2 9 ) 2 )( 1 ) 1 ( 3 ( 2 ) 2 ( 2 1 ) 1 ( 1 + + + + =- + +- =- + + +- =- k k k i i k k i 1 1 ) 2 )( 1 ( ) 2 ( ) 2 ( + = + =- + +- =- k k i i k i i k i i 1 1 ) 2 )( 1 ( 9 ) 2 )( 1 3 ( 2 + +- + +- + +- = k k k k 9 ) 2 )( 1 ( 9 9 ) 2 )( 1 3 ( 9 2 1 1 + +- + +- +-- = k k k k 9 )) 1 ( 9 ( ) 1 3 (( ) 2 ( 9 2 1 +- +--- = + k k k 9 ) 9 9 1 3 ( ) 2 ( 9 2 1-- +--- = + k k k 9 ) 8 6 ( ) 2 ( 9 2 1----- = + k k 9 ) 4 3 )( 2 ( ) 2 ( 9 2 1 +---- = + k k 9 ) 4 3 ( ) 2 ( 9 2 2 +...
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COT3100Hmk04Sol - COT 3100 Fall 2002 Homework #4 Solutions...

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