What is the nth term of the sequence2,6,12,20,?
Answers
Given the sequence
2
,
6
,
12
,
20
, write it out in a line with gaps between each term:
2
00000
6
00000
12
00000
20
Add a line of term below listing the differences between each pair of terms:
2
00000
6
00000
12
00000
20
000
4
00000
6
000000
8
Add another line of term below listing the differences between each pair of terms:
2
00000
6
00000
12
00000
20
000
4
00000
6
000000
8
000000
2
000000
2
Notice that both terms of the last line are the same.
That implies that our given sequence can be matched using a quadratic formula, which we can construct using the first term of each of the lines as a coefficient:
a
n
=
2
0
!
+
4
1
!
(
n
−
1
)
+
2
2
!
(
n
−
1
)
(
n
−
2
)
a
n
=
2
+
4
n
−
4
+
n
2
−
3
n
+
2
a
n
=
n
2
+
n
If you know the formulas for
N
∑
n
=
1
n
2
and
N
∑
n
=
1
n
then you can just add them to provide the formula for
N
∑
n
=
1
a
n
.
I would rather construct it directly:
Add another line to the top of our sequences, consisting of the sums of the terms on the original top line:
0
00000
2
00000
8
00000
20
00000
40
000
2
00000
6
00000
12
00000
20
000000
4
00000
6
000000
8
000000000
2
000000
2
We want to start with
s
1
=
2
. So ignore the first terms of each sequence and use the second terms to give us our formula for the sum to
n
terms:
s
n
=
2
0
!
+
6
1
!
(
n
−
1
)
+
6
2
!
(
n
−
1
)
(
n
−
2
)
+
2
3
!
(
n
−
1
)
(
n
−
2
)
(
n
−
3
)
s
n
=
2
+
6
n
−
6
+
3
n
2
−
9
n
+
6
+
1
3
n
3
−
2
n
2
+
11
3
n
−
2
s
n
=
1
3
(
n
3
+
3
n
2
+
2
n
)
s
n
=
1
3
n
(
n
+
1
)
(
n
+
2
)