f r 2 6 r 9 r 3 2 So a n c 1 3 n c 2 n 3 n We must have c 1 3 and 3 c 1 3 c 2 3

F r 2 6 r 9 r 3 2 so a n c 1 3 n c 2 n 3 n we must

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f. r 2 + 6 r + 9 = ( r + 3) 2 . So a n = c 1 ( - 3) n + c 2 n ( - 3) n . We must have c 1 = 3 and - 3 c 1 - 3 c 2 = - 3 so c 2 = - 2. a n = (3 - 2 n )( - 3) n . g. r 2 + 4 r - 5 = ( r - 1)( r + 5) . So a n = c 1 + c 2 ( - 5) n . We must have c 1 + c 2 = 2 and c 1 - 5 c 2 = 8 so c 1 = 3 and c 2 = - 1. a n = 3 - ( - 5) n . 8. a. L n = L n - 1 + L n - 2 2 . b. The characteristic equation of the recurrence is r 2 - 1 2 r - 1 2 = 1 2 (2 r 2 - r - 1) = 1 2 (2 r + 1)( r - 1). This has roots - 1 / 2 and 1. So L n = 1 ( - 1 / 2) n + 2 (1 n ) = 1 ( - 1 / 2) n + 2 . Using 100000 = L 1 = - 1 / 2 + 2 and 300000 = L 2 = 1 / 4 + 2 we get 1 = 800000 3 and 2 = 700000 3 . Therefore L n = 800000 3 ( - 1 2 ) n + 700000 3 .
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12. The characteristic equation of a n = 2 a n - 1 + a n - 2 - 2 a n - 3 is r 3 - 2 r 2 - r + 2 = ( r - 1)( r - 2)( r + 1), which has roots 1 , 2 , - 1. Therefore we can write a n = 1 · 1 n + 2 · 2 n + 3 · ( - 1) n . Using the initial conditions, we have 3 = a 0 = 1 + 2 + 3 6 = a 1 = 1 + 2 2 - 3 0 = a 2 = 1 + 4 2 + 3 . We solve this, to get 1 = 6 , 2 = - 1 , 3 = - 2. Therefore a n = 6 - 2 n - 2( - 1) n . § 8.4 30. Let G ( x ) = P k 0 a k x k . a. Then P k 2 a k x k = 2 G ( x ). b. a 0 x + a 1 x 2 + a 2 x 3 + · · · = 2 G ( x ). c. a 2 x 4 + a 3 x 5 + a 4 x 6 + · · · = x 2 ( a 2 x 2 + a 3 x 3 + a 4 x 4 + . . . ) = x 2 ( G ( x ) - a 0 - a 1 x ) . d. a 2 + a 3 x + a 4 x 2 + · · · = 1 x 2 ( a 2 x 2 + a 3 x 3 + a 4 x 4 + . . . ) = 1 x 2 ( G ( x ) - a 0 - a 1 x ) . e. If G ( x ) = ( a 0 + a 1 x + a 2 x 2 + a 3 x 3 + . . . ) then G 0 ( x ) = ( a 1 + 2 a 2 x + 3 a 3 x 2 + . . . ) . f. G ( x ) 2 = ( a 0 + a 1 x + a 2 x 2 + a 3 x 3 + . . . ) 2 = ( a 2 0 + 2 a 0 a 1 x + (2 a 0 a 2 + a 2 1 ) x 2 + (2 a 0 a 3 + 2 a 1 a 2 ) x 3 + . . . ) . 32. Use generating functions to solve the recurrence relation a k = 7 a k - 1 with the initial condition a 0 = 5. Let G ( x ) = P k 0 a k x k . From the recurrence, we have a k x k = 7 a k - 1 x k .
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