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**Unformatted text preview: **oceeding as
before, replacing in this case the variable k with the appropriate whole number, beginning
(−1)0
(−1)1
(−1)2
(−1)3
1
with 0, we get b0 = 2(0)+1 = 1, b1 = 2(1)+1 = − 3 , b2 = 2(2)+1 = 1 and b3 = 2(3)+1 = − 1 .
5
7
(This sequence is called an alternating sequence since the signs alternate between + and −.
The reader is encouraged to think what component of the formula is producing this eﬀect.) 9.1 Sequences 553 3. From {2n − 1}∞ , we have that an = 2n − 1, n ≥ 1. We get a1 = 1, a2 = 3, a3 = 5 and
n=1
a4 = 7. (The ﬁrst four terms are the ﬁrst four odd natural numbers. The reader is encouraged
to examine whether or not this pattern continues indeﬁnitely.)
4. Proceeding as in the previous problem, we set aj =
1
a4 = 2 and a5 = 0. 1+(−1)j
,
j j ≥ 2. We ﬁnd a2 = 1, a3 = 0, 5. To obtain the terms of this sequence, we start with a1 = 7 and use the equation an+1 = 2 − an
for n ≥ 1 to generate successive terms. When n = 1, this equation becomes a1...

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