PASCAL TRIANGLE & FIBONACCI NUMBERS

 

                                PASCAL  TRIANGLE

                                                   1  

                                           1              1

                                   1              2             1

                            1              3              3                1

                    1             4               6             4                   1

          1               5             10            10                5                  1

1                6             15             20             15                 6                     1

  PASCAL TRIANGLE GIVES THE BINOMIAL 

COEFFICIENT IN THE EXPANSION OF BINOMIAL THEOREM.

READ THE BLOG ON "INDIAN RAILWAY "

FOR Example 

 1)  In the expansion of  (a+b)¹ = a + b = 1a+ 1b

   The coefficient of a & b is  1  &  1 .

  The Coefficient of 1st term & 2 nd term is 1 & 1 i.e.  1       1

This is shown in the second row of Pascal's Triangle .

2) In the  expansion of (a+b)² = a² +2ab+b² =1a² +2ab+1b² 

  The Coefficient of 1st term is 1 , Coefficient of 2nd term is 2 & Coefficient of 3rd term is 1 i.e.  1      2      1

This is shown in the 3rd row of Pascal's Triangle.

3)In the expansion of  (a+b)³ = a³ +3a²b+3ab²+b³

                                                  = 1a³ +3a²b+3ab²+1b³

The Coefficient of 1st term is 1, coefficient of 2nd term is 3, coefficient of 3rd term is 3 & Coefficient of last term is 1. i.e.

The Coefficient are   1    3    3    1

This is shown in the 4th row of Pascal's Triangle.

4) Same way expansion of of (a+b)⁴

  (a+b)⁴= a⁴ +a³b¹+a²b²+a¹b³+b⁴=

             = 1a⁴+4a³b¹+6a²b²+4a¹b³+1b⁴

The Coefficients are    1    4     6    4    1 

This is shown in the 5th row of Pascal's Triangle . 

This way Next row in Pascal triangle shows 

the Coefficients of (a+b)⁵ i.e. 1    5   10    10    5    1  & so on .

NOTE : 1) IN THE THIRD ROW OF PASCAL TRIANGLE 2

IS OBTAIN BY  ADDING NUMBER IN PREVIOS ROW i.e. 1+1

2)Fourth row is obtain from 3rd row. i.e. 3 is obtain from 1+2 & next 3 is obtain from 2+1.

3)simillarly fifth row is obtain from 4th row i.e. 4 is obtain from1+3 , 6is obtain from 3+3 , next 4is obtain from 3+1  & so on.

THE FIBONACCI SEQUENCE IS ALSO OBTAIN FROM

PASCAL'S TRIANGLE.

The Sequence is  1      1      2       3        5        8        13       21        34      55        89        144       233      377       610       987        1497  .......  so on

READ THE BLOG ON "BINOMIAL THEOREM"

             

In Fibonacci Sequence  1st & 2nd term is 1 i.e.   sequence is  1     1

3rd term is  1+1 =2  i.e now the sequence is   1    1    2

4th term is  1+2=3 i.e now the sequence is      1    1    2     3 

5th term is  2+3=5 i.e now the sequence is  1   1    2    3    5 

6th term is 3+5=8 i.e. now the sequence is 1    1    2     3      5    8

7thet erm is 5+8=13 i.e. now the sequence is 1  1   2   3    5     8    13

8the term is 8+13=21  , 9th term 13+21=34  ,  10 th term  21+34=55 &  so on .