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201Smt2_1061 - n 0(1 x)n = n 1 n 2 x x2 n n xn(1 xn 1 = 1 x...

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(1 + x ) n = n 0 + n 1 x + n 2 x 2 + · · · + n n x n (1 - x n +1 ) (1 - x ) = 1 + x + x 2 + x 3 + · · · + x n 1 (1 - x ) = 1 + x + x 2 + x 3 · · · = i =0 x i 1 (1 + x ) n = - n 0 + - n 1 x + - n 2 x 2 + · · · = i =0 - n i x i = 1 + ( - 1) n + 1 - 1 i x + ( - 1) 2 n + 2 - 1 2 x 2 + · · · = i =0 ( - 1) i n + i - 1 i x i 1 (1 - x ) n = - n 0 + - n 1 ( - x ) + - n 2 ( - x ) 2 + · · · = i =0 - n i ( - x ) i = 1 + ( - 1) n + 1 - 1 i ( - x ) + ( - 1) 2 n + 2 - 1 2 ( - x ) 2 + · · · = i =0 n + i - 1 i x i 1
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MACM 201 Test 2 March 8, 2006. 50 minutes Total marks: 55. Marks are indicated by ( ). (1) (15) Consider the following recurrence relation; a n +6 a n - 1 +14 a n - 2 +16 a n - 3 +8 a n - 4 = 0 , a 0 = - 1 , a 1 = 1 , a 2 = 2 , a 3 = 3 Write down the general solution and the equations that will determine the unknown coefficients, but do not solve for the coefficients . 2
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(2) (10) Consider the following recurrence relation; a n - 2 a n - 1 - 3 a n - 2 = 2(3 n ) , a 0 = 2 , a 1 = 3 (a) Use the recurrence relation to determine a 2 and a 3 . (b) Solve the recurrence relation. 3
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(c) Calculate a 2 and a 3 using your answer from (b). 4
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(3) (10) Solve the following recurrence relation using generating functions; a n +2 + 3 a n + 1 = n 3 + 2 n , a 0 = 1 , a 2 = 2 5
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(4) (10) For n 1 let
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