deep study on pascal triangle

“PASCAL TRIANGLE”

Pascal was not only the scientist, but also great mathematician . His efforts towards science and math were countless; Pascal triangle is a triangle upon which still some works has to done. I had also done some work on Pascal triangle; there came some points that made his invention . I studied some more on Pascal triangle then there some points came in front of me that i want to share those points with you.

Firstly we going to learn that, How this triangle is build?

If we add numbers i.e.(1+1=2) (1+2+1=4) (1+3+3+1=8) (1+4+6+4+1=16) we get (2, 4, 8, 16) ,which are even numbers.

Further reading tells, 121 is square of 11, 1331 – 11³, 14641 – (11²)², but for the solution of 11 raise to the power 5, new methods has been developed.

By using this method, we can solve any power of 11. This triangle can easily be remembered. Just remember 11, in the next number 1-1 is at the both end and just add the middle number. For example 11, 1, 2, 1, 1, (1+2), (2+1), 1 => 1, 3, 3, 1, 1, (1+3), (3+3), (3+1), 1 => 1, 4, 6, 4, 1.

Figure of Pascal Triangle

11

2
2

1
2
1

4
4

1
3

3
1

8
8

1
4

6

4
1

16
7

1
5

10

10

5
1

32
5

1
6

15

20

15

6
1

64
1

1
7

21

35

35

21

7
1

128
2

1
8

28

56

70

56

28

8
1

256
4

1
9

36

84

126

126

84

36

9
1

512
8

1
10

45

120

210

252

210

120

45

10
1

1024
7

1
11

55

165

330

462

462

330

165

55

11
1

2048
5

1
12

66

220

495

792

924

792

495

220

66

12
1
4096
1

11
11
8192
2

11²
121
16384
4

11³
1331
32768
8

11
14641
65536
7

11
161051
131072
5

11
1771561
262144
1

11
19487171
524288
2

11
214358881
1048576
4

11
2357947691
2097152
8

11
25937424601
4194304
7

11
2.85312E+11
8388608
5

11
3.13843E+12
16777216
1

33554432
2

67108864
4

134217728
8

268435456
7

536870912
5

1073741824
1

2147483648
2

4294967296
4

8589934592
8

17179869184
7

34359736368
5

115 -> 1 5 10 10 5 1 -> which is a form pascal’s triangle.

To write this in 115 from we have to use this format .

100000

50000

10000

1000

50

1

One is at sixth place .we will add five zero after it . Five is at fifth place we will add four zero .Like this we keep on decreasing number of zero…….

Add at the end we will find out sum of all values and that will be the value of (115).

161051

Is value of 115

Like this we will find out value of (1110) .

First through pascal’s triangle we get :

1 10 45 120 210 252 210 120 45 10 1 form . To

find out value we will use this format.

10000000000

10000000000

4500000000

1200000000

210000000

25200000

2100000

120000

4500

10

1

Add ten zero after 1.

Add nine zero after 10.

After 45 add 8 zero.

After 120 add 7 zero.

After 210 add 6 zero.

After 252 add 5 zero.

After 210 add 4 zero.

After 120 add 3 zero.

After 45 add 2 zero.

Add 1 zero after 10.

At the end add1.

Thisis 1110 value 25937424601

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