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Unformatted text preview: 70 266 2016 = v u with the additional property that 360 + v u . 4. (9 marks) A sequence { } n x is defined by 11 1 = x , 23 2 = x and 2 1 12 + = n n n x x x for 3 n . Prove that ( ) n n n x 3 4 2 = for all P n . 5. (8 marks) Prove by induction that n n 1 2 1 ... 3 1 2 1 1 2 2 2 + + + + for all positive integers n . You may use, without proof, the fact that ( ) 1 1 1 1 1 2 + + k k k for 1 k . 6. (a) (5 marks) Suppose Z c b a , , . Prove that if a = b + c , then gcd( a , b ) = gcd( b , c ). (You must prove this directly without using Proposition 2.21, which states that if Z r q b a , , , with a = qb + r , then gcd( a , b ) = gcd( b , r ).) (b) (4 marks) Suppose P n . Determine, with proof, all possible values of gcd(8 n + 20, 5 n + 13). Be sure to also show by example that each of these values is possible....
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This note was uploaded on 10/21/2010 for the course MATH 135 taught by Professor Andrewchilds during the Fall '08 term at Waterloo.
 Fall '08
 ANDREWCHILDS

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