Answer:
a
n
=
4
⋅
2
n
−
1
if it is a geometric sequence.
or could be:
a
n
=
1
3
(
2
n
3
−
6
n
2
+
16
n
)
if not.
Explanation:
There is a common ratio between successive pairs of terms:
8
4
=
2
16
8
=
2
32
16
=
2
So this looks like a geometric sequence with initial term
a
=
4
and common ratio
r
=
2
.
If so, the formula for the
n
th term is:
a
n
=
a
r
n
−
1
=
4
⋅
2
n
−
1
This is probably the answer expected by the questioner.
However, note that any finite sequence of terms does not determine an infinite sequence - unless you are told what kind of sequence it is - e.g. arithmetic, geometric, harmonic.
For example, we can match these first
4
terms with a cubic formula as follows:
Write down the sequence as a list:
4
,
8
,
16
,
32
Write down the sequence of differences between each pair of terms:
4
,
8
,
16
Write down the sequence of differences of this sequence:
4
,
8
Write down the sequence of differences of this sequence:
4
Having reached a constant sequence (albeit consisting of only one term), we can use the initial term of each of the sequences we have found as coefficients of a formula for the
n
th term:
a
n
=
4
0
!
+
4
1
!
(
n
−
1
)
+
4
2
!
(
n
−
1
)
(
n
−
2
)
+
4
3
!
(
n
−
1
)
(
n
−
2
)
(
n
−
3
)
=
4
+
(
4
n
−
4
)
+
(
2
n
2
−
6
n
+
4
)
+
(
2
3
n
3
−
4
n
2
+
22
3
n
−
4
)
=
2
3
n
3
−
2
n
2
+
16
3
n
=
1
3
(
2
n
3
−
6
n
2
+
16
n
Step-by-step explanation:
