x3solnfall2009 - 0 ·· 3 k k-1 3 k 1 k 1-1 = 3 ...

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CONCORDIA UNIVERSITY COMP 232/2 Mathematics for Computer Science FALL 2009 Examination 3 1. Solve the following system of congruences: 2 x + 5 y 1 (mod 8) , 3 x + 7 y 1 (mod 8) . 2. Use the Euclidean algorithm to find gcd(761 , 531). Show all your steps. 3. Use mathematical induction to prove that 3 · 1 · 0 + 3 · 2 · 1 + ··· + 3 · n · ( n - 1) = n 3 - n for every positive integer n . 4. The Fibonacci numbers f 0 , f 1 , ..., are defined by f 0 = 0 , f 1 = 1 , f n = f n - 2 + f n - 1 for n 2 . Apply the facts that f k +4 = f k +2 + f k +3 and f k +2 = f k + f k +1 for all k 0 to construct a proof by induction that f n f n +3 - f n +1 f n +2 = ( - 1) n +1 for all n 0. Solutions 1. Since (modulo 8) we have ± 2 · x + 5 · y 1 3 · x + 7 · y 1 ² ⇐⇒ ± 2 · x + 5 · y 1 1 · x + 2 · y 0 ² ⇐⇒ ± 0 · x + 1 · y 1 1 · x + 2 · y 0 ² ⇐⇒ ± 0 · x + 1 · y 1 1 · x + 0 · y ≡ - 2 6 ² ⇐⇒ ± x 6 (mod 8) y 1 (mod 8) ² it follows that x ∈ { 8 n + 6 | n Z } , y ∈ { 8 n + 1 | n Z } . 2. n q n r n 0 761 1 531 2 1 230 3 2 71 4 3 17 5 4 3 6 5 2 7 1 1 = gcd 8 2 0 3. Basis, n = 1 : 3 · 1 · 0 + ··· + 3 · n · ( n - 1) = 3 · 1 · (1 - 1) = 3 · 1 · 0 = 0 = 1 3 - 1 = n 3 - n . Assume: 3 · 1 · 0 + ··· + 3 · k · ( k - 1) = k 3 - k for some k 1. Then: 3 · 1 ·
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Unformatted text preview: 0 + ··· + 3 · k · ( k-1) + 3 · ( k + 1) · (( k + 1)-1) = 3 · 1 · 0 + ··· + 3 · k · ( k-1) + 3 · ( k + 1) · k = k 3-k + 3 · ( k + 1) · k = k 3-k + 3 k 2 + 3 k = ( k 3 + 3 k 2 + 3 k + 1)-( k + 1) = ( k + 1) 3-( k + 1) . 4. Basis, n = 0 : f n f n +3-f n +1 f n +2 = f f 3-f 1 f 2 = 0 · 2-1 · 1 =-1 = (-1) 0+1 = (-1) n +1 . Assume: f k f k +3-f k +1 f k +2 = (-1) k +1 for some k ≥ 0. Then: f ( k +1) f ( k +1)+3-f ( k +1)+1 f ( k +1)+2 = f k +1 f k +4-f k +2 f k +3 = f k +1 ( f k +2 + f k +3 )-( f k + f k +1 ) f k +3 = f k +1 f k +2 + f k +1 f k +3-f k f k +3-f k +1 f k +3 = f k +1 f k +2-f k f k +3 =-( f k f k +3-f k +1 f k +2 ) =-(-1) k +1 = (-1) k +2 = (-1) ( k +1)+1 . 2-Dec-2009, 10:43 pm...
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This note was uploaded on 11/15/2010 for the course ENCS COMP 232 taught by Professor Ford during the Fall '10 term at Concordia University Irvine.

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