Pascal’s Triangle
One of the most interesting number
patterns in mathematics is Pascal’s triangle. The triangle at first was first
extensively studied by Blaise Pascal, though we find many ancient Indian
scholars and those from china and Persia also studied it.
To build the triangle, let us start
with 1;
On the second, write two 1s, altogether
forming three corners of triangle.
Now on each subsequent row start and
end with 1’s and compute each interior term by summing two numbers above it.
If we observe the
diagonals of triangle, the first diagonal is of 1’s, next diagonal has counting
numbers, and the third has triangular numbers. (1, 3, 6, 10, 15 . . .)
Interesting
facts and features about Pascal’s Triangle
1. Powers of 2
If we sum each row, we obtain power of base two starting
with;
1 = 20
1+1 = 21
1+2+1 = 22
1+3+3+1 = 23
1+4+6+4+1 = 24
And so on…
2. Powers of 11
The triangle also reveals power of base 11. All you have to
do is squish the number in each row together.
1 = 110
11 = 111
121 = 112
1331 = 113
14641 = 114
But when you reach to fifth row, you get double entries (two
digits number, ex: 10).
In that case, we proceed as;
Now,
161051 = 115 and so on…
We can locate the perfect squares of natural numbers from the second
diagonal. The square of a number is equal to the sum of the numbers next to it
and below of this number.
3.
Expanding binomials or
to find formulas for (x+y)n
We can use Pascal’s
triangle to find binomial expansions for (x+y)n or to find the
formulas for higher powers like; (x+y)4 ,(x+y)5, etc.
For this, we proceed
as follows;
In case of (x+y)2,
the coefficient is found in the second row (see the figure1 for the notations
of rows).
|
1x2
+
|
2xy+
|
1y2
|
For (x+y)3,
we go to the third row.
|
1x3+
|
3x2y+
|
3xy2+
|
1y3
|
For (x+y)4,
we look into fourth row.
|
1x4 +
|
4x3y
+
|
6x2y2
+
|
4xy3
+
|
1y4
|
By this you can find the expansion for
any power of binomial expression.
4.
Last but not the
least!
We can split
the Pascal’s triangle into two halves, it is symmetric! The separated parts are
identical.
0 comments: