Discrete Mathematics with Graph Theory (3rd Edition) 152

# Discrete Mathematics with Graph Theory (3rd Edition) 152 -...

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150 Solutions to Exercises (d) The generating function for this geometric sequence is 1~"" as in PAUSE 10. (e) The generating function is 3 + 3x + 3x 2 + 3x 3 + ... = 3(1 + x + x 2 + ... ) = 1~"" (f) [BB] The generating function is 1 + x 2 + x4 + x 6 + ... = 1 + x 2 + (x 2 )2 + (x 2 )3 + .. '. This is obtained by replacing x by x 2 in (12). The generating function is 1-':",2' (g) The generating function is 1 - 2x + 3x 2 - 4x 3 + .. '. This is obtained by replacing x by -x in (13). The generating function is (1':",)2' 3. [BB] f(x) = ao + a1X + a2x2 + ... + anx n + .. . 2xf(x) = 2aox + 2a1x2 + ... + 2an_1Xn + .. . Subtracting gives f(x) - 2xf(x) = ao + (a1 - 2ao)x + ... + (an - 2an_1)X n + ... = 1 since ao = 1 and an - 2an -1 = 0 for n ~ 1. Thus, f(x) _1_ = 1 + 2x + (2X)2 + ... + (2x)n + ... 1-2x 1 + 2x + 4x 2 + ... + 2 n x n + ... We conclude that an = 2n. Our solution is the same as the sequence defined in (5) of Section 5.2 except that it has one additional term, ao = 1. 4. We have f(x) = 3xf(x) = Subtracting gives ao + a1X + a2 x2 + 3aox + 3a1X2 + f(x)
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