Question

Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the nth month?

Answer 1

Answer:

There will be as many pairs as the value of the nth term of the Fibonacci sequence

Step-by-step explanation:

During a month, only those pairs of rabbits that were given birth the previous month dont produce their pair. For the month n you take the amount of pairs of the n-1 month (lets call it an-1) and we have to add the new pairs created, that were created for rabits given birth on the month n-2 or before (in other words, the active ones). This means that an-2 pairs were created, so the total number of pairs, an, is given by the formula

$a_{n} = a_{n-1} + a_{n-2}$ (n has to be greater than 2)

or, equivalently

$a_{n+2} = a_{n+1} + a_{n}$

This is Fibonacci's formula for values greater than 2, also note that

a1 = 1

a2 = 1 ( because the pair was inactive this month)

a3 = 2 (because now the pair is activa)

As a result, we have

a1 = 1

a2 = 1

an+2 = an+1 - an (with n at least 1)

Thus, there were as many pairs in the nth month as the value of the nth term in the Fibonacci sequence.