Assn5Soln - p 2 the property that NI appears twice, etc....

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1. Explain the relationship between the solutions of a non-homogeneous linear recurrence relation and its homogeneous version. Let f 1 ( n ) and f 2 ( n ) be solutions to a non-homogeneous linear recur- rence relation (A) k i =0 c i a n - i = r ( n ) and g ( n ) be a solution to the homogeneous version of this recurrence, (B) k i =0 c i a n - i = 0. Then f 1 ( n ) - f 2 ( n ) is a solution to (B), and f 1 ( n )+ g ( n ) is a solution to (A). 2. Solve the following recurrences: a n = - 4 a n - 1 - 4 a n - 2 , a 0 = 3 , a 1 = 5 a n +3 = 8 a n +2 + 18 a n - 21 a n +1 , a 0 = 0 , a 1 = 7 , a 2 = 41 The general solution is a n = ( αn + β )( - 2) n , and taking the initial conditions into account, we get a n = ( - 11 2 n + 3)( - 2) n . The general solution is a n = α 2 n + ( βn + γ )3 n , giving a n = - 2 n + (2 n + 1)3 n , 3. Evaluate (1 + 3 i ) 12 . Solve the recurrence a n = 2 a n - 1 - 4 a n - 2 . (1 + 3 i ) 12 = (2(cos 60 + i sin 60)) 12 = 2 12 (cos 720 + i sin 720) = = 4096(cos 0 + i sin 0) = 4096 a n = α sin(60 n ) + β cos(60 n ). 4. Chapter 8.1, exercise 7. Note that the only pairs that can repeat are IN, NI, IO, OI, NO and ON. Let p 1 be the property that IN appears twice,
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Unformatted text preview: p 2 the property that NI appears twice, etc. Then N ( p 1 ) = N ( p 2 ) = . . . = N ( p 6 ) = 9! / 4, N ( p 1 p 4 ) = N ( p 1 p 5 ) = N ( p 2 p 3 ) = N ( p 2 p 6 ) = N ( p 3 p 6 ) = N ( p 4 p 5 ) = 7! / 2, and no other combinations are possible. Hence, there are 11! 8-6 9! 4 + 6 7! 2 such words. 5. Chapter 10.2, exercises 9 and 16. 10.2.9: a n = a n-1 + a n-2 , a = a 1 = 1, a n = 1 5 p 1 + 5 2 P n +1-p 1- 5 2 P n +1 10.2.16: a n = a n-1 + a n-2 the last number in the sum is either two (then the rest sums to n-2), or greater than two (then we can decrease it by one, getting an expression that sums to n-1). a 1 = 0, a 2 = 1, a n = 1 5 p 1 + 5 2 P n-1-p 1- 5 2 P n-1...
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This note was uploaded on 07/15/2010 for the course MACM 201 taught by Professor Marnimishna during the Spring '09 term at Simon Fraser.

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Assn5Soln - p 2 the property that NI appears twice, etc....

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