The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about a.d. 1200 as a model for the growth of rabbit populations. It is given by the recurrence relation f_{n + 1} = fn + f_{n - 1}, for n = 1, 2, 3,? . . ., where f₀ = 1, f₁ = 1. Each term of the sequence is the sum of its two predecessors.?

a. Write out the first ten terms of the sequence.

b. Is the sequence bounded?

c. Estimate or determine ?the ratio of the successive terms of the sequence. Provide evidence that ? = (1 + ?5/2), a number known as the golden mean.?

d. Use induction to verify the remarkable result that?

Transcribed Image Text:

|| 9- lim n→∞ fn+1 fn fn = 1 V5 = (¢" − (−1)"¢¯"). -