Math Dude 016 • Video Extra! • The Fibonacci Sequence...and Rabbits!

Summary

In this video, Jason Marshall explains Fibonacci's Rabbit Problem, where two newborn rabbits are placed in a pen, and the goal is to determine how many rabbits will be in the pen after one year. The problem illustrates the Fibonacci sequence, with each pair of rabbits producing offspring each month. The sequence grows rapidly as rabbits reproduce, starting with 1 pair in month 1, reaching 144 pairs by the 12th month, and 233 pairs by the 13th. The video contrasts Fibonacci's growth model with geometric growth, where doubling the population each month leads to an exponential increase, ending with 8,192 rabbits after a year.

Takeaways

• 😀 Fibonacci posed a problem about how many rabbits will be in a pen after one year, given certain assumptions about their reproduction rate.
• 🐇 Rabbits are assumed to always produce one male and one female offspring, and they can start reproducing at one month old.
• 📅 The rabbit population starts with one newborn pair at the beginning of the first month, which is too young to reproduce.
• 🗓️ By the second month, the original pair has matured and produces one new pair of rabbits, bringing the total to two pairs.
• 🔄 From the third month onward, each pair of adult rabbits mates, producing new pairs of offspring each month.
• 📊 The number of rabbit pairs follows the Fibonacci sequence: each number is the sum of the two previous numbers.
• 🔢 The sequence starts as 1, 1, 2, 3, 5, 8, and continues, with each month's total being the sum of the previous two.
• 📈 By the 12th month, the Fibonacci number reaches 144 pairs of rabbits, but this is the number at the beginning of the 12th month, not the end.
• 📅 To get the total number of pairs at the end of one year, you need to look at the beginning of the 13th month, which is 233 pairs (466 rabbits).
• 📉 In contrast to the Fibonacci model, if the rabbit population grew exponentially (doubling each month), there would be 8,192 rabbits at the end of one year.
• 💡 The Fibonacci sequence demonstrates slower, more controlled growth compared to the rapid, geometric growth of a population doubling each month.

Q & A

What is Fibonacci's Rabbit problem about?

Fibonacci's Rabbit problem asks how many rabbits would be in a pen after one year if two newborn rabbits are placed in the pen, assuming they reproduce under specific conditions such as producing one male and one female offspring each month, starting reproduction when they are one month old, and never dying.

What assumptions does Fibonacci make in his rabbit problem?

Fibonacci assumes that rabbits reproduce once per month, that each pair produces one male and one female offspring, that rabbits start reproducing when they are one month old, and that they never die.

What is the pattern of the rabbit population growth in Fibonacci's problem?

The pattern of rabbit population growth follows the Fibonacci sequence, where each number is the sum of the two previous numbers. The sequence starts with 1, 1, and continues as 2, 3, 5, 8, and so on.

How do we calculate the number of rabbit pairs at the beginning of each month?

At the beginning of each month, the number of rabbit pairs is the sum of the previous two months' totals. Initially, there's one pair, and as they mature, they begin to reproduce, increasing the number of pairs each month.

What is the Fibonacci sequence, and how does it relate to the rabbit population?

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. In the case of the rabbit population, the number of rabbit pairs at the beginning of each month follows this sequence.

How many pairs of rabbits will there be at the beginning of the 13th month?

At the beginning of the 13th month, there will be 233 pairs of rabbits, or 466 individual rabbits.

Why is the total number of rabbit pairs at the beginning of the 12th month not the correct answer?

The number of rabbit pairs at the beginning of the 12th month is not the final total for the year because the question asks for the number of pairs at the end of the 12th month. The correct answer requires the number of pairs at the beginning of the 13th month, which is 233 pairs.

How does the rabbit population growth compare to geometric growth?

Fibonacci's rabbit population grows according to the Fibonacci sequence, which is slower than geometric growth. In geometric growth, the population would double every month. If the rabbit population grew geometrically, there would be 8,192 rabbits at the end of the year, a much larger number than the Fibonacci model's 466 rabbits.

What would happen if the rabbit population followed geometric growth instead of the Fibonacci sequence?

If the rabbit population followed geometric growth, where it doubled every month, there would be 8,192 rabbits at the end of the year, instead of just 466 in the Fibonacci model.

How many rabbit pairs are there at the beginning of the 5th month?

At the beginning of the 5th month, there are 8 pairs of rabbits, as indicated by the Fibonacci sequence: 1, 1, 2, 3, 5, 8.