Discrete Mathematics with Graph Theory (3rd Edition) 157

# Discrete Mathematics with Graph Theory (3rd Edition) 157 -...

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Section 5.4 155 12. [BB] f(x) ao + aIX + a2x2 + a3x3 + aox + aIx 2 + a2x3 + aox 2 + aIx 3 + aox 3 + + anx n + + an_IX n + + an_2 Xn + + an_3 Xn + xf(x) x 2 f(x) x 3 f(x) Therefore, f(x) - xf(x) - x 2 f(x) + x 3 f(x) = ao + (al - ao)x + (a2 - al - ao)x 2 + (a3 - a2 - al + ao)x 3 + ... + (an - an-I - an-2 + an_3)X n + ... = 2 - 3x+2x 2 since ao = 2, al = -1, a2 = 3 and an - an-I - an-2 + an-3 = 0 for n 2: 3. So and f(x) = 2 - 3x + 2X2 1- x - x 2 + x 3 2 - 3x + 2x2 A th h· . ( ) 2( ). s e mt suggests, we wnte I-x l+x obtaining 2 - 3x + 2x2 Ax + B C f(x) = (1 _ x)2(1 + x) = (1 - x)2 + 1 + x (Ax+B)(I+x)+C(I-x)2 = 2-3x+2x2 (B + C) + (A + B - 2C)x + (A + C)x 2 = 2 - 3x + 2x2 So B + C = 2, A + B - 2C = -3, A + C = 2. The solution to these equations is A = B = i, C = ~. Therefore, I( I+X) 7( 1 ) f(x) 4 (l-x)2 +4 l+x i(1 + x)(1 + 2x + 3x 2 + ... + (n + l)xn + ... ) +Hl - x + x 2 - x 3 + ... + (_I)nxn + ... ) i (1 + 2x + 3x 2 + 4x 3 + ... ) + i (x + 2x2 + 3x 3 + ... ) + ~ (1 -
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Unformatted text preview: x + x 2 -x 3 + ... ) i(1 + 3x + 5x 2 + 7x 3 + ... + (2n + l)xn + ... ) + ~(1 -x + x 2 -x 3 + ... ) Thus, an = i(2n + 1) + ~(_I)n. 13. (a) The characteristic polynomial of the Pell sequence is x 2 -2x -1 whose roots are x = 1 ± .J2. Thus Pn = CI (1 + .J2)n + c2(1 -.J2)n for constants CI and C2. Since Po = 1 and PI = 2, we have CI + C2 = 1 and CI (1 + .J2) + c2(1 -.J2) = 2, so CI = 2+/'2 and C2 = 2-4 v'2. Thus Pn = (2+4v'2) (1 + .J2)n + (2-40) (1 -.J2)n. (b) Note that 11 -.J21 < ~, so 11 -.J2l n < ~ for all n 2: 1. Also.J2 < 2 < 2 + .J2 implies o < 2 -.J2 < 2, so 0 < 2-4 v'2 < ~ = ~. Thus the second term in the formula for Pn obtained in Exercise 13(a) is less than ~ for any n, that is, that IPn -(2+4v'2) (1 + .J2)nl < ~ for any n. Since Pn is an integer, this part follows from the fact that there is precisely one integer within ~ of any given real number....
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## This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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