My teacher said the pascal's triangle was really cool, and has a lot of patterns.
What are they?
There are myriads of patterns to the Pascal's triangle. Here is the basic Pascal's triangle. To begin, let's just look at the numbers is each row.
We can already see that the sum of the numbers in each row is a power of 2.
\(1=2^0\\ 1+1=2^1\\ 1+2+1=2^2\\ 1+3+3+1=2^3\\ ...\)
Futhermore, just looking at the digits, they are powers of 11!
\(1=11^0\\ 11=11^1\\ 121=11^2\\ 1331=11^3\\ 14641=11^4\\\)
However, when we get to the next row, we need to advance the ten's digit.
Then when we look down the diagonal's, from the top, we see:
- The first row is all ones: 1, 1, 1, ...
- The second row is counting numbers: 1, 2, 3, ...
- The third row are triangular numbers: 1, 3, 6, ...
- The fourth row are tetrahedral numbers: 1, 4, 10, ...
From the diagram above, we see the Fibonacci Numbers as the sum of each row.
Some more patterns:
-The numbers in the triangle can be represented by combination.
-If you shade all the odd numbers, you see a beautiful fractal.
These are just some patterns, you can see all of them on this amazing website: https://www.cut-the-knot.org/arithmetic/combinatorics/PascalTriangleProperties.shtml.
The Pascal's triangle was one of the most beautiful mathematical creations ever made. There are so many arrangments and identities derived from a simple pattern.
I hope this helped,