Answer:
Recursive formulation:
{
a
1
=
−
4
a
n
+
1
=
5
a
n
000
(
n
=
1
,
2
,
3
,
...
)
Explanation:
We are given:
{
a
1
=
−
4
a
4
=
−
500
The general formula for the
n
th term of a geometric series is:
a
n
=
a
r
n
−
1
where
a
is the initial term and
r
is the common ratio.
A recursive formula can be given as:
{
a
1
=
a
a
n
+
1
=
r
a
n
000
(
n
=
1
,
2
,
3
,
...
)
In our example:
5
3
=
125
=
−
500
−
4
=
r
4
r
1
=
a
r
4
−
1
a
r
1
−
1
=
r
3
So the only possible Real value for
r
is
3
√
5
3
=
5
.
Footnote
There are two other possibilities for a geometric sequence with
a
1
=
−
4
and
a
4
=
−
500
, which are sequences of Complex numbers.
This is because
5
3
has two other cube roots, namely
5
ω
and
5
ω
2
, where
ω
=
−
1
2
+
√
3
2
i
is the primitive Complex cube root of
1
. Either of these will also work as a suitable common ratio.
