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Math Tutorial Series

Arithmetic Progression

Higher Order (Part 2)

Un : Pascal Triangle

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Musical

English Subtitle

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Arithmetic Progression

Higher Order

Method 1 : Pascal Triangle

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Leonardo Sembiring

leonardo.sembiring@gmail.com

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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

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arithmetic progression order 2

a0

a0

a1

a1 + a0

a0

a1 + 2a0

a1 + 3a0

a0

......

a1

a2

a2 + a1

a1 + a0

a0

a2 + 2a1+ a0

a1 + 2a0

a2 + 3a1+ 3a0

a1 + 3a0

a0

a2 + 4a1+ 6a0

Arithmetic Progression Order 2

......

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a2

a1

a0

U1

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U2

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U3

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U4

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U5

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1

1

1

1

1

2

1

3

3

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4

6

1

1

1

1

1

1

1

1

1

1

1

3

2

4

3

1

4

5

6

1

5

10

10

1

1

6

6

15

15

21

21

35

35

7

7

20

1

6

4

3

3

2

1

1

1

1

1

1

Pascal Triangle

add

put

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...

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...

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...

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...

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...

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n-1C0

n-1C1

n-1C2

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...

n-1Cn-3

n-1Cn-2

n-1Cn-1

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nC0

nC1

nC2

...

​

...

nCn-2

nCn-1

nCn

Pascal Triangle

can also be written as follows...

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Un

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...

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n-1C0

n-1C1

n-1C2

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...

n-1Cn-3

n-1Cn-2

n-1Cn-1

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nC0

nC1

nC2

...

​

...

nCn-2

nCn-1

nCn

a2

a1

U1

U2

U3

U4

U5

​

Un

a0

a2

a2 + a1

a2 + 2a1+ a0

1.

U1

=

=

a2

a2

1.

U2

=

=

a1

+

1.

a2

1.

U3

=

a1

+

2.

a0

+

1.

=

.a2

=

.a1

+

.a0

+

n-1C1

n-1C0

n-1C2

Un formula

and so on..

Arithmetic Progression Order 2

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(n-1-0)!

(n-1)!

=

n-1C0

(n-1)!

(n-1)!

=

0!

1

=

(n-1-1)!

(n-1)!

=

n-1C1

1!

(n-2)!

(n-1)!

=

1!

n-1

=

(n-2)!

(n-1)(n-2)!

=

1!

(n-1-2)!

(n-1)!

=

n-1C2

2!

(n-3)!

(n-1)!

=

2!

(n-1)(n-2)

=

(n-3)!

(n-1)(n-2)(n-3)!

=

2!

2

0! = 1

0!

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Un

.a2

=

.a1

+

.a0

+

n-1C1

n-1C0

n-1C2

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n-1C0

a2

a1

a0

n-1C1

n-1C2

1

(n - 1)

1/2(n - 1)(n - 2)

Un

.a2

=

.a1

+

.a0

+

(n-1)

1

(n - 1)(n - 2)

Un

a2

=

.a1

+

.a0

+

(n-1)

(n - 1)(n - 2)

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Un formula for the example...

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2

a0

a1

a2

33

22

13

11

9

7

2

2

6

5

1

Un

a2

=

.a1

+

.a0

+

(n-1)

(n - 1)(n - 2)

Un

1

=

.5

+

.2

+

(n-1)

(n - 1)(n - 2)

1

=

+

+

5n - 5

n2 - 3n + 2

=

n2 + 2n - 2

Un

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Un formula in general...

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a0

a3

a2

a1

U1

U2

U3

U4

U5

​

Un

Un formula

and so on..

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n-1C0

n-1C1

n-1C2

​

...

n-1Cn-3

n-1Cn-2

n-1Cn-1

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nC0

nC1

nC2

...

​

...

nCn-2

nCn-1

nCn

arithmetic progression order 1

arithmetic progression order 2

arithmetic progression order 3

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maximum index 1

maximum index 2

maximum index 3

maximum index 4

n-1C0

n-1C1

n-1C2

Arithmetic Progression Order

×

×

×

n-1C3

n-1C4

×

×

and so on...

1

2

3

4

+

+

+

+

Un

=

a1

a0

a2

a1

a0

a3

a2

a1

a0

a4

a3

a2

a1

a0

fixed index (n-1)

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SMART IDEAS FOR MATHEMATICS

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