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Unformatted text preview: e nth term of
a given sequence, and it is only through experience gained from evaluating sequences from explicit
formulas that we learn to begin to recognize number patterns. The pattern 1, 4, 9, 16, . . . is rather
recognizable as the squares, so the formula an = n2 , n ≥ 1 may not be too hard to determine.
With this in mind, it’s possible to see 2, 5, 10, 17, . . . as the sequence 1 + 1, 4 + 1, 9 + 1, 16 + 1, . . .,
so that an = n2 + 1, n ≥ 1. Of course, since we are given only a small sample of the sequence, we
shouldn’t be too disappointed to ﬁnd out this isn’t the only formula which generates this sequence.
For example, consider the sequence deﬁned by bn = − 4 n4 + 2 n3 − 31 n2 + 25 n − 5, n ≥ 1. The
reader is encouraged to verify that it also produces the terms 2, 5, 10, 17. In fact, it can be shown
that given any ﬁnite sample of a sequence, there are inﬁnitely many explicit formulas all of which
generate those same ﬁnite points. This means that there will be inﬁnitely many correct answe...
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