Question

The Fibonacci sequence is named after an Italian mathematician: Leonardo Bonacci (c. 1170-c. 1250)-known as Fibonacci, and also Leonardo of Pisa, Leonardo Pisano Bigollo, and as Leonardo Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics. In mathematical terms, the sequence F of Fibonacci numbers is defined by the recurrence relation FF1F-2 with seed values F1, F21or F1-0, F2-1 The Liber Abaci began the sequence with F, 1 The Fibonacci sequence is represented by the numbers in the following integer sequence: 1,1, 2,3,5, 8, 13, 21,34,55,89,144,... or (often, in modern usage): 0,1,1, 2,3,5, 8,13,21,34,55,89,144,. By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two. Write a MATLAB script which computes the Fibonacci sequence and plots the ratio of successive members of the series (e.g. plot 1/1 1;2/1 2; 3/2-1.5;5/3-1.67; etc.). Show that this converges to a single value, the "golden ratio" of 1.61803398875 (etc.). How many iterations does it take before successive iterations do not change by more than 0.000,001 or 10 or 1E-6?

Answer 1

n = 2;
while ratio>0;
 ratio = fibonacci(n)./fibonacci(n-1);
 ratio1 = fibonacci(n+1)./fibonacci(n);
 plot(n,ratio,'--o')
 hold on
 if abs(ratio -ratio1)<=0.000001;
  fprintf('%i\n', x)
  break;
 end
 i=i+1;
end

line(xlim,[1.618 1.618])
hold off