# Fibonacci Sequence to 72nd term and Multiplication Fibonacci Powers

Fibonacci de Pisa was the most influential 12th century western mathematician. His father was an important Italian merchant or trader of goods. On a trip to Egypt around 1200AD, Fibonacci was introduced to Arabic numerals (0,1,2,3,4,5,6,7,8,9….) formerly unknown to Europe where the system of numbers were Roman (I,II,III,IV,V,VI,VII,VIII,IX,X…..). The Arabic numerals changed the world forever as it opened up whole new doors for mathematicians, scientists, and merchants.

Fibonacci published a book in 1202 called Liber Abaci (Book of Calculation) that introduced Arabic numerals and a sequence of numbers he discovered by simply starting with the numbers 0 and 1. Every number in the sequence is found by adding the previous two numbers. So, the first numbers of this sequence are

0,1,2,3,5,8,13,21,34,55,89,144…..

The numbers grow at a rapid rate. For example, the 100th number in the sequence is 354,224,848,179,261,915,075 (354 quintillion ….).

As I challenged the children to build these numbers, I disclosed that on my first attempt at generating these numbers at their age, I made a mistake on the 19th number in the sequence: 4,181 by only one digit (I mistakenly failed to add a regrouping carry). Consequently, every successive number was wrong. Upon learning of my mistake by the 50th number, I found the error and recreated the next 31 numbers and then continued to the 500th Fibonacci number (it has 75 digits):

139,423,224,561,697,880,139,724,382,870,407,283,950,070,256,587,697,307,264,108,962,948,325,571,622,863,290,691,557,658,876,222,521,294,125

I created a scroll of these numbers and since every 4 or 5 numbers adds one place value, the numbers resemble a right triangle.

The attached pdf allows them to generate these numbers and check their answers along the way. There is no shame in making a mistake; but it would be a shame not to try to find your mistake and continue on.

KEY STRATEGY: The only way I have found to avoid mistakes is to line up the place values as well as the answers and carries (regrouping). Always circle your carries to avoid mistaking these numbers for sums. This best practice takes a lot of concentration and discipline. I use this method personally today and rarely mis-calculate my bank deposits and tax calculations.

NEW DISCOVERY: shared with 3rd graders and beyond

I made a new discovery this week after studying Fibonacci numbers for the last 40+ years. I haven't found anything published about this yet, but I am sure it is out there. I tried to multiply the Fibonacci numbers and failed at 0 and 1 because every number following 1 was a zero. I failed at 1 and 1 since every number following the second 1 was a one. But them when I started with 1 and 2, it followed with:

1 2 2 4 8 32 256 8192 2,097,152

Then I noticed that these all are powers of 2:

1=2^0 2=2^1 2=2^1 4=2^2 8=2^3 32=2^5 256=2^8 8192=2^13 2,097,152=2^21

Then, I noticed that each exponent or power are in fact the original Fibonacci sequence. The reason is that when you multiply equivalent bases, the product is that base to the sum of the exponents. This is called the product property of exponents: b^m x b^n = b^(m+n).

An inquisitive fourth grader, Lila, asked what would happen if you did this starting with 1 and 3, or 1 and 4, essentially, bases beyond base 2. We explored this, and sure enough, the exponents of base three were Fibonacci numbers (1,3,9,27,243…). This proves that one discovery in mathematics leads to dozens of new questions, leading to more discovery.

I hope the children create their Fibonacci scroll and continue this throughout the next 8 weeks and into the summer.

AttachmentSize
Fibonacci_to_72nd_and_beyond.pdf1.17 MB
Fibonacci_Multiplication_Powers.pdf681.27 KB