Some sequences are defined by a recursion formula —that is, a formula that defines each term of the sequence in terms of one or more of the preceding terms. For example, if { a n } is defined by a 1 = 1 a n d a n = 2 a n − 1 + 1 f o r n ≥ 2 then a 2 = 2 a 1 + 1 = 2 ⋅ 1 + 1 = 3 a 3 = 2 a 2 + 1 = 2 ⋅ 3 + 1 = 7 a 4 = 2 a 3 + 1 = 2 ⋅ 7 + 1 = 15 and so on. In Problems 63–66, write the first five terms of each sequence. 66. a 1 = 1 and a n = − 1 3 a n − 1 for n ≥ 2
Some sequences are defined by a recursion formula —that is, a formula that defines each term of the sequence in terms of one or more of the preceding terms. For example, if { a n } is defined by a 1 = 1 a n d a n = 2 a n − 1 + 1 f o r n ≥ 2 then a 2 = 2 a 1 + 1 = 2 ⋅ 1 + 1 = 3 a 3 = 2 a 2 + 1 = 2 ⋅ 3 + 1 = 7 a 4 = 2 a 3 + 1 = 2 ⋅ 7 + 1 = 15 and so on. In Problems 63–66, write the first five terms of each sequence. 66. a 1 = 1 and a n = − 1 3 a n − 1 for n ≥ 2
Solution Summary: The author calculates the term for n=2 as follows.
Some sequences are defined by arecursion formula—that is, a formula that defines each term of the sequence in terms of one or more of the preceding terms. For example, if {an} is defined by
a
1
=
1
a
n
d
a
n
=
2
a
n
−
1
+
1
f
o
r
n
≥
2
then
a
2
=
2
a
1
+
1
=
2
⋅
1
+
1
=
3
a
3
=
2
a
2
+
1
=
2
⋅
3
+
1
=
7
a
4
=
2
a
3
+
1
=
2
⋅
7
+
1
=
15
and so on. In Problems 63–66, write the first five terms of each sequence.
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