At the beginning, there was a couple of rabbits (one male and one female) in the farm.

A female rabbit would give birth to one male and one female rabbits monthly.

In the next month, the young couple would give birth to one male and one female rabbits too.

1:2:3:

4: 5……………………….

After n months….

How many pairs of rabbits are in the farm?

Firstly!

For every natural number (1, 2, 3, 4,.....) Let a be the number of the rabbits in the farm at the beginning of the nth month[or the end of the (n-1)th month].

Then !

a =1 a =2 a = sum of the pairs of rabbits in the beginning of the second month a and the first pair of rabbits at the beginning of the first month a [i.e., in the 2nd month, the number of baby pairs a ] = = 3 pairs

a = sum up the pairs of rabbits in the beginning of the third month a and, he first pair of rabbits at the beginning of the second month a [i.e., in the 3rd month, the number of baby pairs a ] =3 + 2 = 5 pairs a : 4 +

a = sum up the pairs of rabbits in the beginning of the (n-1) month a(n-1) and the first pair of rabbits at the beginning of the (n-2) month a(n-2) [ i.e., in the (n-1)th month, the number of baby pairs a ] a = a + a Therefore! a = a + a n a :

Solve (1) and (2) simultaneously,

Month (n) Number of old rabbits Number of baby rabbits Total number of rabbits Let’s see the data !

At last!

We can substitute the number of months (n) in to the equation a n = ( ) n+1 - ( ) n+1. to know how many pairs of rabbits in the farm!

!!BYE!!

Done by : Chu Shun Leung 6C(13) Done by : Chu Shun Leung 6C(13)