# The Fibonacci Sequence

## What is the Fibonacci sequence?

The Fibonacci sequence is the series of numbers who are the sum of the two numbers that proceed it. The first few terms in the sequence are 0, 1, 1 ,2, 3, 5, 8, 13, 21, 34 and so on. As you can see, 0+1=1, 1+1=2, 1+2=3, 2+3=5, 3+5=8... The numbers continue on infinitely. The series can be represented as a rule in terms of xn = xn-1 + xn-2.

## Who was Fibonacci?

Fibonacci was a man known in his time as Leonardo of Pisa. He was a merchant who developed a love for numbers at a young age. The first appearance of the Fibonacci sequence was in his Liber abaci. He discovered the ratio while studying rabbits and the pattern at which they reproduced.

## What is the Golden Ratio and Spiral?

"A Fibonacci spiral is a series of connected quarter-circles drawn inside an array of squares with Fibonacci numbers for dimensions. The squares fit perfectly together because of the nature of the sequence, where the next number is equal to the sum of the two before it. Any two successive Fibonacci numbers have a ratio very close to the Golden Ratio, which is roughly 1.618034. The larger the pair of Fibonacci numbers, the closer the approximation. The spiral and resulting rectangle are known as the Golden Rectangle" (livescience.com).
How to make a Golden Rectangle and Golden Spiral

## Where can it be found in our world?

The Fibonacci sequence, Golden Ratio, Golden Rectangle and Fibonacci Spiral are represented in nature all around us, from the way flowers grow, to the painting "The Last Supper," to spiral galaxies.
The Fibonacci Sequence, The Golden Rectangle and Architecture

## How does second difference apply to the Fibonacci series?

When second difference is applied, and in this case it goes al the way down to seventh difference, an interesting pattern is revealed. The top line of numbers is the first 15 terms of the sequence, and the row below shows the difference between each paris of terms. The seventh difference of the first 15 numbers in the series has ended as a backwards Fibonacci sequence with alternating negative numbers.

## What is Pascal's triangle?

Pascal's triangle is a pyramid of numbers, starting at 1, where the sum of adjacent numbers is the term in the subsequent row, placed underneath and in between the two numbers. It is named after Blaise Pascal, a french philosopher and mathematician.

## How is Pascal's triangle related to the Fibonacci sequence?

If a pattern is created by highlighting numbers by going up and over, as shown in the picture below, the sum of highlighted numbers is the fibonacci sequence.

## What's so special about Pascal's triangle?

Pascal's triangle has many hidden patterns. The sums of its rows follow the pattern of doubling such that each row is 2 to the x power, with row one being 0. The triangle also relates to polynomials, where (x+1) raised to each successive power and foiled has coefficients of the numbers of each row of the triangle. Another hidden gem in Pascal's triangle is the relation of the numbers in each row to the powers of 11. For example, 11 raised to the 3rd power is 1331, which are the terms in the 4 row of the triangle. Pascal's triangle also involves probability and combinations (watch the video below for more).
Binomial Distribution Part 2 Probability and Pascal

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