The Day our Minds were Blown...(swish)

Hello everyone, my name is Kadeem and i was chosen to be the scribe for today by Patrick ( =0... that's my shocked and angry face). And I'm going to work tonight so i don't have very much time so i will start this, go to work, come back and finish.

First, Mr. K promised and reassured us that he was going to blow our minds and show us one way the universe was built, but before that he explained the meaning of "Tedious" to Jojo and whoever else that didn't know the meaning.
Tedious: Tiresome by reason of length, slowness, or dullness; boring.


Binomial Theorem

Mr. K told us to expand and simplify the following:
(a+b)⁰
(a+b)¹
(a+b)²
(a+b)³
(a+b)⁴
(a+b)⁵
(a+b)⁶


After solving the first 5 questions simply using the FOIL method or the short way, he then showed us the PASCAL's Triangle...













The pattern is shown in the red arrows/lines, you must find the sum of the two numbers in the previous row to get the numbers in the next row.

Okay, not watch this carefully...

In this slide you can see the the Pascal's triangle in just basic binomials being raised to an exponent. That is only one of the many patterns in these binomials that make it a lot easier to solve Rather than going through the whole FOIL thing.

Patterns in...
The Coefficient: Adding the two coefficients from the row above.(seen in the Pascal's Triangle)
A: Decreasing by (a ^{n - 1} ), (a ^{n - 2} ), (a ^{n - 3} )...
B: Increasing by (b ^{n + 1} ), (b ^{n + 2} ), (b ^{n + 3} )...

Later on, Mr. K gave us the same triangle and told us to find patterns in the triangle , and we found a number of different ones:
Regular counting numbers
The Sum of Counting numbers a.k.a. "triangular numbers"
Sum of Triangular numbers
powers of 11

Fibonacci Sequence

Leonardo Fibonacci was an French/Italian mathematician...More Info.

the sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... (you get the next number by finding the sum of the two previous numbers). As you can see this sequence appears in Pascal's Triangle and in a lot of the natural living things on earth such as plants, bee's, flowers, even in a Leonardo Da Vinci painting. He was also explaining how "Phi" was in the sequence and how it's in everything from 4x6 pictures and our body's to what we consider beautiful, and then the bell rang...

The next scribe will be....

i haven't decided yet i will get back to you... Good Nite!