Interesting Integer Sequences and Their Stories
Lunzhi Shi
I was first introduced to the Pascal’s Triangle back in year 5 when I was asked to produce a program that loops and gives a constant output of the triangle. As I began to learn more mathematics, I realized that the triangle was far more than an array as you see it.
If you aren’t sure about what Pascal’s Triangle is, it’s like the food triangle: the further down the more there is. One number in the array is simply the sum of the two above it, taking empty spaces as zero and the initial as 1: it keeps on going non-stop. Now over with the definition, let’s first take a look at the applications of the triangle. I will start off with this easy one related to everyday algebra: Try expanding this: ( + ) . Now this: ( + ) 2 . That will be 2 + 2 + 2 . How about ( + ) 3 ? It will get more and more complicated.
But why am I mentioning binomial expansion? Is that somehow related to the triangle? Have a look at the rearranged table of the expansions (up to the power of 5):
Row Expansion 0
1 +
1 2 3 4 5
1 x
1 y
1 x 2
1 y 2
+
2 xy
+
1 x 3
3 x 2 y
3 xy 2
1 y 3
+
+
+
1 x 4
4 x 3 y
6 x 2 y 2
4 xy 3
1 y 4
+
+
+
+
1 x 5
5 x 4 y
10 x 3 y 2
10 x 2 y 3
5 xy 4
1 y 5
+
+
+
+
+
As you can see, the coefficients draw out the first five rows of the Pascal’s Triangle. Say we have ( + ) , you then need to find row to get the coefficient. The row and the expansion are magically related. The powers also come in with a very clear pattern of adding up and subtracting down to zero. This fun fact leads to so called binomial theorem. To further explain this, you possibly know that we need to use ( ) or ( choose ). This is combinatorics; representing the number of ways of picking
3
Made with FlippingBook - professional solution for displaying marketing and sales documents online