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**Unformatted text preview: **ellipsis, . . ., indicate that the proposed pattern continues forever. Each of
the numbers in the list is called a term, and we call 1 the ‘ﬁrst term’, − 3 the ‘second term’, 9 the
2
4
8
‘third term’ and so forth. In numbering them this way, we are setting up a function, which we’ll
call a per tradition, between the natural numbers and the terms in the sequence.
1 Recall that this is the set {1, 2, 3, . . .}. 552 Sequences and the Binomial Theorem n a(n) 1 1
2
−3
4
9
8
27
− 16 2
3
4
.
.
. .
.
. In other words, a(n) is the nth term in the sequence. We formalize these ideas in our deﬁnition of
a sequence and introduce some accompanying notation.
Definition 9.1. A sequence is a function a whose domain is the natural numbers. The value
a(n) is often written as an and is called the nth term of the sequence. The sequence itself is
usually denoted using the notation: an , n ≥ 1 or the notation: {an }∞ .
n=1
Applying the notation provided in Deﬁnition 9.1 to the sequence given (1), we have a1 = 1 , a2 = − 3 ,
2
4
a3 = 9 and so forth....

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