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M135W09A3

# M135W09A3 - MATH 135 Assignment#3 Hand-In Problems 1 In...

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Unformatted text preview: MATH 135 Assignment #3 Hand-In Problems 1. In each part, state the answer. (a) Determine the value of (b) Determine the value of (c) Determine the value of 7 . 0 13 . 10 7 . 2 Winter 2009 Due: Wednesday 28 January 2009, 8:20 a.m. 2. Write down the expansion of (2x2 + y )5. 3. Determine the coeﬃcient of x9 in the expansion of (2x3 − 3x)5 . 4. Determine the coeﬃcient of x in the expansion of 5. Express 42 43 44 x (y − 1)4 x (y − 1)2 + x+ 2 1 0 4 4 x(y − 1)6 + (y − 1)8 3 4 as the fourth power of a polynomial in x and y . 6. Prove that for every positive integer n n n n +···+ + n 1 0 and (b) n n n − + − 2 1 0 + (−1)n−1 n n + (−1)n n n−1 =0 7 3 2x − x 2 8 . (a) = 2n 7. The Fibonacci sequence is a sequence of positive integers f1 , f2 , · · · deﬁned recursively by f1 = f2 = 1 and fn = fn−1 + fn−2 for all n 3. n (a) Prove that r =1 2 fr = fn fn+1 for all n ∈ P. (b) Prove that fn+1 < ( 7 )n for all n ∈ P. 4 8. A sequence of integers x1 , x2 , · · · is deﬁned recursively by x1 = −1, x2 = 2 xn = 4xn−2 for all n 3. Prove that xn = (−1)n 2n−1. ...continued Recommended Problems 1. Text, page 105, #28 2. Text, page 108, #63 3. Text, page 106, #37 4. Text, page 106, #42 5. Determine the coeﬃcient of z 41 in the expansion of (z 2 − 2z 5 )10 . 6. In the expansion of (1 − x2 )(1 + x)2n , the coeﬃcient of x2 is 189. Determine n. ...
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