•
The elements of the sequence are given.
•
A formula to generate the terms of the sequence is given.
•
A recursive formula to generate the terms of the sequence is given.
Example 2
The examples below illustrate sequences given by listing all the el-
ements,
1.
{
1
,
5
,
10
,
4
,
−
98
,
1000
,
0
,
2
,...
}
.
2.
±
1
,
1
2
,
1
3
²
. Though the elements are listed, we can also guess a formula
to generate them, what is it?
3.
{−
1
,
1
,
−
1
,
1
,
−
1
,
1
}
. Though the elements are listed, we can also guess
a formula to generate them, what is it?
Example 3
The examples below illustrate sequences given by a simple formula.
Notice that
n
does not have to start at
1
.
1.
±
1
n
²
∞
n
=1
. The elements of the sequence are:
±
1
,
1
2
,
1
3
²
. The general
term of the sequence is
a
n
=
1
n
.
2.
³
√
n
−
3
´
∞
n
=3
. The elements are
³
0
,
√
1
,
√
2
,
√
3
´
. In this case,
n
could
not start at
1
.
3.
{
(
−
1)
n
}
∞
n
=2
. The elements of the sequence are
{
1
,
−
1
,
1
,
−
1
}
.
Example 4
The examples below illustrate sequences given by a recursive for-
mula.
1.
±
a
1
=1
a
n
=2
a
n
−
1
+5
. We can use this formula to generate all the terms.
But the terms have to be generated in order. For example, in order to get
a
10
,weneedtoknow
a
9
, and so on. Using the formula, we get that
a
2
a
1
=7
Having found
a
2
, we can now generate
a
3
a
3
a
2
9
Andsoon
.