Part 3/6:
- Month 3 ( ( a_3 ) ): The original pair reproduces, resulting in a total of two pairs, thus ( a_3 = 2 ).
For a month ( n ) where ( n \geq 3 ), the growth can be expressed as:
[
a_n = a_{n-1} + a_{n-2}
]
This recursive relation results from the fact that the total number of pairs in any month is equal to the number of pairs from the previous month plus the number of pairs from two months earlier (which are the pairs that have become mature enough to reproduce).
By continuing this process, we can generate terms of the sequence:
( a_4 = a_3 + a_2 = 2 + 1 = 3 )
( a_5 = a_4 + a_3 = 3 + 2 = 5 )
( a_6 = a_5 + a_4 = 5 + 3 = 8 )
The sequence continues as ( 1, 1, 2, 3, 5, 8, ... ), which is precisely the Fibonacci sequence, defined recursively as:
[