Fibonnaci numbers

by

John R. Simmons

Who was Fibonacci?

Fibonacci was born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD. He was considered the greatest European mathematician of th middle ages. His full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa. He introduced the decimal number system ito Europe.

The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, ...(add the last two numbers to get the next). They are defined recursively by the formula f1=1, f2=1, fn= fn-1 + fn-2 for n>=3.

We will derive a formula for the sum of the first n fibonacci numbers and prove it by induction.

n = 1 2 3 4 5 6 7 8 9 10 11 12...

 n = 1 2 3 4 5 6 7 8 9 10 fn = 1 1 2 3 5 8 13 21 34 55 sum fn = 1 2 4 7 12 20 33 54

Notice from the table it appears that the sum of the first n terms is the (nth+2) term minus 1. We will use mathematical induction to prove that in fact this is the correct formula to determine the sum of the first n terms of the Fibonacci sequence.

Click here to see proof by induction

Next we will investigate the sum of the squares of the first n fibonacci numbers.

 n = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 fn 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 fn^2 1 1 4 9 25 64 169 441 1156 3025 7921 sum fn^2 1 2 6 15 40 104 273 714 1870 4895 12816

Notice from the table it appears that the sum of the squares of the first n terms is the nth term multiplied by the (nth+1) term . We can use mathematical induction to prove that in fact this is the correct formula to determine the sum of the squares of the first n terms of the Fibonacci sequence.

Fibonacci Numbers and Nature

On Monday, April 25, 2005. NASA and European Space Agency (ESA) released new views of one of the most well-known image Hubble has ever taken, spiral galaxy M51 known as the Whirlpool Galaxy.

NASA/ESA, Hubble Whirlpool Galaxy

You can go to my Essay, "Fibonacci Numbers in Nature" to see a discussion of the Hubble Whirlpool Galaxy.

The Mathematical Magic of the Fibonacci Numbers