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M135F08A3 - MATH 135 Assignment#3 Hand-In Problems 1 In...

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MATH 135 Fall 2008 Assignment #3 Due: Wednesday 01 October 2008, 8:20 a.m. Hand-In Problems 1. In each part, state the answer. (a) Determine the value of 18 0 . (b) Determine the value of 19 17 . (c) Determine the value of 9 3 . (d) Determine the value of 12 5 . (e) Write down the expansion of ( x 2 - 2 y ) 4 . (f) Determine the coefficient of x 9 in the expansion of (2 x 3 - 5) 5 . 2. A sequence of integers is defined by x 1 = 9, x 2 = 3, x m +2 = - x m +1 + 6 x m for m 1. Prove that x n = 3 · 2 n - ( - 3) n for all n P . 3. A sequence of integers is defined by y 1 = 0, y 2 = 4, y m = 4 y m - 2 for m 3. Prove that y n = 1 2 (2 n + ( - 2) n ) for all n P . 4. A sequence of integers is defined by t 1 = 3, t 2 = 7, t 3 = 33, t r = 7 t r - 2 + 6 t r - 3 for r 4. Prove that t n = 2( - 1) n - ( - 2) n + 3 n for all n P . 5. Simplify 5 0 x 5 + 5 1 x 4 (2 - x )+ 5 2 x 3 (2 - x ) 2 + 5 3 x 2 (2 - x ) 3 + 5 4 x (2 - x ) 4 + 5 5 (2 - x ) 5 . Show your work. 6. Determine the coefficient of x 7 in the expansion of 2 x 2 - 3 x 8 . 7. Prove that n r r s = n s n - s r - s . 8. Define f ( x ) = (1 + x ) n . In each part, as always, be sure to justify your answer. (a) Determine the value of a so that f ( a ) = n 0 + n 1 + n 2 + · · · + n n - 1 + n n . (b) Determine the value of n 0 + n 1 + n 2 + · · · + n n - 1 + n n . (c) Determine the value of n 0 - n 1 + n 2 - · · · + ( - 1) n - 1 n n - 1 + ( - 1) n
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