Unformatted text preview: quence of numerators and denominators, respectively, we have an = dn . After
n
some experimentation,5 we choose to write the ﬁrst term as a fraction and associate the
4
negatives ‘−’ with the numerators. This yields 1 , −2 , 13 , −8 , . . .. The numerators form the
17
19
sequence 1, −2, 4, −8, . . . which is geometric with a = 1 and r = −2, so we get cn = (−2)n−1 ,
for n ≥ 1. The denominators 1, 7, 13, 19, . . . form an arithmetic sequence with a = 1 and
d = 6. Hence, we get dn = 1 + 6(n − 1) = 6n − 5, for n ≥ 1. We obtain our formula for
n−1
c
an = dn = (−2)−5 , for n ≥ 1. We leave it to the reader to show that this checks out.
6n
n
While the last problem in Example 9.1.3 was neither geometric nor arithmetic, it did resolve into
a combination of these two kinds of sequences. If handed the sequence 2, 5, 10, 17, . . ., we would
be hardpressed to ﬁnd a formula for an if we restrict our attention to these two archetypes. We
said before that there is no general algorithm for ﬁnding the explicit formula for th...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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