Exercise 4: Fibonacci's Original Rabbit Reproduction Sequence (and the Golden Ratio)

In this video I go over the first appearance of the famous Fibonacci sequence and show that the limit of the ratio of two consecutive terms is equal to the Golden Ratio. The Italian mathematician Leonardo Bonacci, or more commonly known as Fibonacci (short for "filius Bonacci or "son of Bonacci"), first wrote the Fibonacci sequence in 1202 when analyzing the population growth of an idealized rabbit population. Assuming rabbits live forever, and starting with a pair of rabbits that reproduce another pair after 2 months of age, the population starts growing according to the Fibonacci sequence:

the current population = the population 1 month ago + the population 2 months ago

I then show that the limit of the ratio of consecutive terms of the sequence, (population at n + 1 month) / population at n month), is equal to the famous golden ratio. I go over the history and more instances of the Fibonacci sequence and the golden ratio in the next video!

#math #sequences #fibonaccisequence #goldenratio #calculus

- Leonardo Bonacci: https://en.wikipedia.org/wiki/Fibonacci
- Fibonacci sequence: https://en.wikipedia.org/wiki/Fibonacci_sequence

Timestamps:

- Exercise 4: Fibonacci problem about rabbit reproduction: 0:00
- Solution to part (a): Recap on the Fibonacci sequence: 0:51
- Solving the rabbit reproduction problem: 1:40
- Writing down the sequence of new rabbit pairs being reproduced: 3:33
- Sequence is just the Fibonacci sequence: 6:07
- Solution to part (b): Ratio of the Fibonacci sequence terms: 6:40
- Rewriting the sequence to plug in the Fibonacci sequence: 7:58
- Proved our sequence recurrence relation: 10:00
- Finding the limit of the sequence, using Theorem 2 (Exercise 1): 10:41
- Solving the limit using quadratic formula: 13:36
- Taking the positive value as our limit: 14:45
- Our limit is actually the Golden Ratio! 15:23

Notes and playlists:

- Summary: https://inleo.io/threads/view/mes/re-leothreads-21jgbnz5x
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUgfK34Voi6BvrNqt4X4BV
- Notes: https://peakd.com/mathematics/@mes/infinite-sequences-limits-squeeze-theorem-fibonacci-sequence-and-golden-ratio-more
- Infinite Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .

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