Chapter 7
Use the following to answer questions 1-16:
In the questions below, describe each sequence recursively. Include initial conditions and
assume that the sequences begin with
a
1
.
1.
a
n
=
5
n
.
Ans:
a
n
=
5
a
n
−
1
,
a
1
=
5.
2. The Fibonacci numbers.
Ans:
a
n
=
a
n
−
1
+
a
n
−
2
,
a
1
=
a
2
=
1.
3. 0
,
1
,
0
,
1
,
0
,
1
,
…
.
Ans:
a
n
=
a
n
−
2
,
a
1
=
0,
a
2
=
1.
4.
a
n
=
1
+
2
+
3
+
...
+
n
.
Ans:
a
n
=
a
n
−
1
+
n
,
a
1
=
1.
5. 3
,
2
,
1
,
0
,−
1
,−
2
,
…
.
Ans:
a
n
=
a
n
−
1
−
1,
a
1
=
3.
6.
a
n
=
n
!
.
Ans:
a
n
=
na
n
−
1
,
a
1
=
1.
7. 1
/
2
,
1
/
3
,
1
/
4
,
1
/
5
,
…
.
Ans:
1
1
1
n
n
n
a
a
−
−
+
a
=
,
a
1
=
1
/
2.
8. 0
.
1, 0
.
11, 0
.
111, 0
.
1111
,
…
.
Ans:
a
n
=
a
n
−
1
+
1
/
10
n
,
a
1
=
0
.
1.
9. 1
2
,
2
2
,
3
3
,
4
2
,
…
.
Ans:
a
n
=
a
n
−
1
+
2
n
−
1,
a
1
=
1.
10. 1
,
111
,
11111
,
1111111
,
…
.
Ans:
a
n
=
100
a
n
−
1
+
11.
11.
a
n
=
the number of subsets of a set of size
n
.
Ans:
a
n
=
2
⋅
a
n
−
1
,
a
1
=
2.
12. 1
,
101
,
10101
,
1010101
,
…
.
Ans:
a
n
=
100
a
n
−
1
+
1,
a
1
=
1.
Page 88