Math Tutorial Series
Arithmetic Progression
Higher Order (Part 5)
Sn : Pascal Triangle
Musical
English Subtitle
Arithmetic Progression
Higher Order
Sn : Pascal Triangle
“
Leonardo Sembiring
leonardo.sembiring@gmail.com
2
arithmetic progression
a0
a1
a2
33
22
13
11
9
7
2
2
6
5
1
subtract
put
=
arithmetic progression order 2
=
Arithmetic Progression Order 2
a0
a1
a1 + a0
a0
a1 + 2a0
a1 + 3a0
a0
......
a1 + 2a0
a0
1
Pascal Triangle
6
15
10
1
5
1
arithmetic progression order 2
a1 + a0
a0
a1 + 3a0
a0
a2 + 4a1+ 6a0
Arithmetic Progression Order 2
......
1
1
1
1
1
1
1
1
3
2
4
3
1
4
6
1
5
10
1
1
6
15
21
21
35
35
7
7
20
| a2 | a1 | a0 |
S1 | | | |
S2 | | | |
S3 | | | |
S4 | | | |
S5 | | | |
1
2
5
10
10
4
4
6
3
1
3
1
a0
a1
| a2 | a1 | a0 |
U1 | | | |
U2 | | | |
U3 | | | |
U4 | | | |
U5 | | | |
1
6
4
3
3
2
1
1
1
1
1
1
add
put
add
put
a2
a2 + a1
a2 + 2a1+ a0
a2 + 3a1+ 3a0
1
1
1
1
1
2
1
3
3
1
4
6
| | | | | | | 1 | | | | | | | |
| | | | | | 1 | | 1 | | | | | | |
| | | | | 1 | | 2 | | 1 | | | | | |
| | | | 1 | | 3 | | 3 | | 1 | | | | |
| | | 1 | | 4 | | 6 | | 4 | | 1 | | | |
| | ... | | ... | | ... | | ... | | ... | | ... | | |
n-1C0 | n-1C1 | n-1C2 | | ... | n-1Cn-3 | n-1Cn-2 | n-1Cn-1 | | ||||||
nC0 | nC1 | nC2 | ... | | ... | nCn-2 | nCn-1 | nCn | ||||||
Pascal Triangle
can also be written as follows...
Sn
.a2
=
.a1
+
.a0
+
nC2
nC1
nC3
| | | | | | | 1 | | | | | | | |
| | | | | | 1 | | 1 | | | | | | |
| | | | | 1 | | 2 | | 1 | | | | | |
| | | | 1 | | 3 | | 3 | | 1 | | | | |
| | | 1 | | 4 | | 6 | | 4 | | 1 | | | |
| | ... | | ... | | ... | | ... | | ... | | ... | | |
n-1C0 | n-1C1 | n-1C2 | | ... | n-1Cn-3 | n-1Cn-2 | n-1Cn-1 | | ||||||
nC0 | nC1 | nC2 | nC3 | | ... | nCn-2 | nCn-1 | nCn | ||||||
a2
a1
a0
a2
2a2 + a1
3a2 + 3a1+ a0
1.
S1
=
=
a2
a2
2.
S2
=
=
a1
+
1.
a2
3.
S3
=
a1
+
3.
a0
+
1.
=
Sn formula
and so on..
Arithmetic Progression Order 2
S1 |
S2 |
S3 |
S4 |
S5 |
|
Sn |
(n-1)!
n!
=
nC1
(n-1)!
n(n-1)!
=
1!
n
=
(n-2)!
n!
=
nC2
2!
(n-2)!
n(n-1)(n-2)!
=
2!
n(n-1)
=
2
(n-3)!
n!
=
nC3
3!
n(n-1)(n-2)
=
(n-3)!
n(n-1)(n-2)(n-3)!
=
3!
6
1!
Sn
.a2
=
.a1
+
.a0
+
nC2
nC1
nC3
| | |
| | |
| | |
| | |
nC1
a2
a1
a0
nC2
nC3
n
1/2 n(n - 1)
1/6 n(n - 1)(n - 2)
Sn
.a2
=
.a1
+
.a0
+
n(n-1)
n
n(n - 1)(n - 2)
Sn formula for the example...
2
a0
a1
a2
33
22
13
11
9
7
2
2
6
5
1
Sn
n.1
=
.5
+
.2
+
n(n-1)
n(n - 1)(n - 2)
n
=
+
+
15n - 15
2.(n2 - 3n + 2)
=
(2n2 + 9n - 5)
Sn
Sn
.a2
=
.a1
+
.a0
+
n(n-1)
n
n(n - 1)(n - 2)
6
n
×3
×3
Sn formula in general...
a0
a3
a2
a1
S1 |
S2 |
S3 |
S4 |
S5 |
|
Sn |
| | | | | | | 1 | | | | | | | |
| | | | | | 1 | | 1 | | | | | | |
| | | | | 1 | | 2 | | 1 | | | | | |
| | | | 1 | | 3 | | 3 | | 1 | | | | |
| | | 1 | | 4 | | 6 | | 4 | | 1 | | | |
| | ... | | ... | | ... | | ... | | ... | | ... | | |
n-1C0 | n-1C1 | n-1C2 | | ... | n-1Cn-3 | n-1Cn-2 | n-1Cn-1 | | ||||||
nC0 | nC1 | nC2 | nC3 | ... | nCn-2 | nCn-1 | nCn | |||||||
arithmetic progression order 1
arithmetic progression order 2
arithmetic progression order 3
Sn formula
and so on..
maximum index 3+1
maximum index 1+1
maximum index 2+1
maximum index 4+1
nC1
nC2
nC3
Arithmetic Progression Order
×
×
×
nC4
nC5
×
×
and so on...
1
2
3
4
+
+
+
+
Sn
=
a1
a0
a2
a1
a0
a3
a2
a1
a0
a4
a3
a2
a1
a0
fixed index n
SMART IDEAS FOR MATHEMATICS
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