at the beginning, there was a couple of rabbits (one male and one female) in the farm
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<Fibonacci Numbers><Fibonacci Numbers>
Maths. ProjectMaths. Project
Fibonacci NumbersFibonacci Numbers
• At the beginning, there was a couple of rabbits (one male and one female) in the farm.
Fibonacci NumbersFibonacci Numbers
• A female rabbit would give birth to one male and one female rabbits monthly.
Fibonacci NumbersFibonacci Numbers
• In the next month, the young couple would give birth to one male and one female rabbits too.
1:1:
2:2:
3:3:
Explanation:Explanation:
4:4:
5……………………….5……………………….
Explanation:Explanation:
After After nn months…. months….
How many pairs of rabbits are in the farm?How many pairs of rabbits are in the farm?
Firstly!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].
n
Fibonacci NumbersFibonacci Numbers
Then !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 ]
1
2
3
1
12
+
+ +
2 + 1
= = 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
43
2
2
a :a :44
+
• 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 ]
• Therefore! a = a + a a = a + a n 1n
2n
2n
n
nna :a :
Calculation:Calculation:
Solve (1) and (2) simultaneously,
Month (n)Month (n) 11 22 33 44 55 66 77 88 99 1010 1111 1212
Number Number of old of old
rabbitsrabbits11 11 22 33 55 88 1313 2121 3434 5555 8989 144144
Number Number of babyof babyrabbitsrabbits
00 11 11 22 33 55 88 1313 2121 3434 5555 8989
Total Total number number of rabbitsof rabbits
11 22 33 55 88 1313 2121 3434 5555 8989 144144 233233
Fibonacci NumbersFibonacci Numbers
Let’s see the data !Let’s see the data !
At last!At last!
We can substitute the number of months (n) in to the equation
an = ( )n+1 - ( )n+1 . 5
12
512
515
1
to know how many pairs of rabbits
in the farm!
!!BYE!!!!BYE!!