154
1.7.
SEQUENCES AND DIFFERENCE EQUATIONS
e.
a
1
= 5
a
n
+1
= 2
a
n
for
n
≥
1
Solution.
a.
Since
n
is the general term, we have 1
,
2
,
3
,
4, and 5 for the first five terms.
b.
For
n
= 1
,
sin
π
2
= 1; for
n
= 2
,
sin
2
π
2
= 0; for
n
= 3
,
sin
3
π
2
=
−
1; for
n
= 4
,
sin
4
π
2
= 0; and for
n
= 5
,
sin
5
π
2
= 1.
c.
Take the first five natural numbers (in order) to find:
1
1+1
=
1
2
,
2
1+2
=
2
3
,
3
1+3
=
3
4
,
4
1+4
=
4
5
, and
5
1+5
=
5
6
d.
Since
π
≈
3
.
141592
· · ·
; we see the first five terms of this sequence is: 1
,
4
,
1
,
5
,
and 9.
e.
This is known as a
recursive
formula because after one (or more) given term(s), the subsequent terms
are found in terms of the given term(s). For this example, the first term is given:
a
1
= 5; for
n
= 2, we
use
a
2
= 2
a
1
= 2(5) = 10; for
n
= 3,
a
3
= 2
a
2
= 2(10) = 20; for
n
= 4,
a
4
= 2
a
3
= 2(20) = 40, and for
n
= 5,
a
5
= 2(40) = 80. In summary, the first five terms of the sequence are 5
,
10
,
20
,
40
,
80.
a50
To visualize a sequence, one can graph the sequence of points
(1
, a
1
)
,
(2
, a
2
)
,
(3
, a
3
)
, . . .