Assn5Soln

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

This preview shows pages 1–2. Sign up to view the full content.

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,

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online