The Fibonacci Sequence works with negative numbers as well.

The numbers alternate in a +- pattern as follows: ...-8,5,-3,2,-1,1,0...

Each number is still the sum of the two previous numbers.

The equation, t =((-1) +1)*t , is used to find Fibonacci Numbers with negative term numbers when the two previous terms are unknown.

Pascal's Triangle

As we saw in class, the sums of the numbers in certain diagonal lines in Pascal's Triangle create the Fibonacci Sequence.

Sample Problems

What is it?

Shockingly, the Fibonacci Sequence is a sequence!

As we learned in Unit 4, a sequence is a list of terms in a pattern.

Starting with term 0, the Fibonacci Sequence is 0,1,1,2,3,5,8,13,21,34,55,89,144...

The numbers in the Fibonacci Sequence are known as Fibonacci Numbers

Fibonacci Sequence and the Golden Ratio

Any number in the Fibonacci Sequence divided by the number prior to it is approximately equal to the Golden Ratio.

The Golden Ratio "φ" is 1.618034...

Where did it come from?

The Fibonacci Sequence was introduced to the Western world by Leonardo Pisano Bigollo.

It was actually known hundreds of years earlier in India.

Fibonacci is also known for introducing the Latin-speaking world to the Hindu-Arabic number system.

**The Fibonacci Sequence**

Where's the pattern?

Using the Golden Ratio

The Golden Ratio can be used to solve for any term in the Fibonacci Sequence using the equation:

t =((φ)-(1-φ) )/√5

This equation will always give an approximate answer.

1. Find the 20th term in the Fibonacci Sequence.

Works Cited

Leonardo Pisano Bigollo

nicknamed Fibonacci meaning "son of Bonaccio"

1170-1250

Portrait of Fibonacci. Digital image. Original Dr. Patrick Flanagan Neurophone. N.p., n.d. Web. 20 Apr. 2014.

**by Caroline Bamberger**

Period 3

Period 3

Every number in the Fibonacci Sequence is equal to the sum of the two numbers before it.

The equation, t =t +t where n=term number and t =the value of the nth term,

can be used to find the next term in the sequence given that n is greater than or equal to 3.

Examples:

3/2=1.5

8/5=1.6

55/34=1.61764705882...

377/233=1.618025751...

A Helpful Pattern

Every nth term in the sequence is a multiple of t such that every 3rd term is a multiple of 2, every 4th term is a multiple of 3 and so on.

The Fibonacci Sequence in Pascal's Triangle. Digital image. Fibonacci. math.rutgers.edu, 1999. Web. 24 Apr. 2014.

The Fibonacci Sequence in Nature

Firstly, the Fibonacci Sequence is commonly found in nature in the form of animal populations. In fact, Fibonacci's original problem displaying the sequence involved the expansion of a rabbit population.

Fibonacci Bunnies

The Fibonacci Sequence represented in the population growth of bunnies. Digital image. Mathfour.com. N.p., n.d. Web. 24 Apr. 2014.

The Golden Spiral

-n

n

n

n

n

(n-1)

(n-2)

n

When squares with widths of the Fibonacci Numbers are organized in a spiral as pictured below, they form what is known as the Golden Spiral.

Diagram of the Golden Spiral. Digital image. Livescience.org. N.p., 14 June 2013. Web. 25 Apr. 2014.

The Golden Spiral in Nature

The Golden Spiral appears commonly in nature in the following ways and many more.

(top left and bottom) Dvorsky, George. The Golden Spiral in nature. Digital image. Io9.com. N.p., 20 Feb. 2013. Web. 25 Apr. 2014.

(top right) Golden Spiral in Human Ear. Digital image. Goldenspiral.net. N.p., 25 Aug. 2012. Web. 25 Apr. 2014.

More Natural Occurrences

The number of petals on a flower tends to be a Fibonacci Number.

Trees branch out in Fibonacci Numbers

The number of spirals of scales on a pine cone or pineapple is typically a Fibonacci Number. Additionally, because different variations of spirals are found on pine cones and pineapples, the numbers of each kind of spiral are adjacent Fibonacci Numbers.

Sunflower seeds also occur in spirals with the same properties as the spirals of scales on pine cones or pineapples.

Fibonacci Numbers in pineapple spirals. Digital image. Kat's Math Blog. Edublogs, 12 Dec. 2012. Web. 25 Apr. 2014.

t =((φ) -(1-φ) )/√5

t =((φ )-(1-φ) )/√5

t =(15127.002-6.61*10 )/√5

t ≈6765

n

20

20

20

2. If t is a Fibonacci number and n = -4, which of the following is true?

A. t = t + t

B. t = t + t

C. t = -(t + t )

D. t = t − t

n

-4

-4

-4

-4

-5

-6

-2

-3

5

6

3

2

2.

t =t +t

t =t +t

t =t +t

So, the answer is A

-4

-4

n

(n-1)

(n-2)

(-4-1)

(-4-2)

-5

-6

Fun Stuff

The 1000th term in the Fibonacci Sequence is 43466557686937456435688527675040625802564660517371780402481729089536555417949051890403879840079255169295922593080322634775209689623239873322471161642996440906533187938298969649928516003704476137795166849228875

A few interesting videos about the Fibonacci Sequence (zoom in)

Puzzles involving the Fibonacci Sequence

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibpuzzles.html#bricks

Asimov, Isaac. "Numbers and Counting." Asimov on Numbers. Garden City, NY: Doubleday, 1977. 47-50. Print.

"Fibonacci Sequence." Fibonacci Sequence. Web. 19 Apr. 2014. <http://www.mathsisfun.com/numbers/fibonacci-sequence.html>.

"Fibonacci sequence." World of Mathematics. Gale, 2007. Gale Power Search. Web. 26 Apr. 2014.

Knott, R. "The Life and Numbers of Fibonacci." Plus.maths.org. 4 Nov. 2013. Web. 20 Apr. 2014. <http://plus.maths.org/content/life-and-numbers-fibonacci>.

Parveen, Nikhat. "Fibonacci in Nature." Fibonacci in Nature. University of Georgia. Web. 19 Apr. 2014. <http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm>.

Reich, Dan. "The Fibonacci Sequence, Spirals, and the Golden Mean." Temple University. Web. 18 Apr. 2014. <https://math.temple.edu/~reich/Fib/fibo.html>.

Schultz, James E., Wade Ellis, Kathleen A. Hollowell, and Paul A. Kennedy. "11.1 Sequences and Series." Algebra 2. Austin, TX: Holt, Rinehart and Winston, 2001. 690-93. Print.

n

n

n

n

n

20

20

-5