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Discrete Mathematics with Graph Theory (3rd Edition) 138

Discrete Mathematics with Graph Theory (3rd Edition) 138 -...

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136 Solutions to Exercises (d) This is the sum of a geometric sequence with a = 1, r = -!. Solving ~ = ar n - 1 , we get n = 61, so the sum is 1- (_!)61 _ ~( ~) 1- (_!) - 3 1 + 261 . (e) This is the sum of a geometric sequence with a = 2, r = 3. Noting that 354294 = 2(3 11 ) and solving ar n - 1 = 354294, we get n = 12. So the sum is 2(1- 3 12 ) = 3 12 _1. 1-3 34. (a) [BB] This is the sum of an arithmetic sequence with a = 1004 and d = -3. With an = -397 = a + (n -l)d, we have 1004 - 3(n -1) = -397, so 3(n -1) = 1401, n -1 = 467, so n = 468. . n 468 The sum IS S = "2[2a + (n -l)d] = 2[2008 + 467( -3)] = 234(607) = 142038. (b) This is the sum of a geometric sequence with a = 324 and r = -~. Solving ar n - 1 - - 131072 gives (_~)n-l - 32768 so n - 1 = 15 n = 16 - 177147 3 - 14348907' ,. The sum is a(l- rn) 324(1- (_~)16) 324(1- (~)16) 3(324)(1- (~)16) 34384948 = 2 = 5 = = 1-r 1-(-'3) '3 5 177147 35. [BB] Suppose the arithmetic sequence ao, ao + d, ao + 2d, . .. is also the geometric sequence aI, al r, al r2, .... Then ao = al. Let's call this number a. Note that one and only one possibility occurs when a = 0 (namely, 0,0,0, ... ) so assume henceforth that a =I- O. Since the sequence is both
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