Some interesting facts about Fibonacci number.

• A question from the lecture about the Fibonacci number is to show that the sum of the first k Fibonacci number equal the k+2 Fibonacci number -1.

We use induction to show it.
Base case:
k = 0, F(0) = 1 Sum(0) = 1 = F(2) -1
Assume k is generic natural number,
   Assume Sum(k) = F(k+2) -1
   Then Sum(k+1)= F(k+2)+F(k+1) -1 = F(k+3) -1
   Then Sum(k+1) = F(k+2+1) -1
Then Sum(k) = F(k+2) -1 Sum(k+1) = F(k+3) -1
Then for all natural number k, Sum(k) = F(k+2)-1
        

• Then looking at a biological fact:

How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?

It turns out the number of pairs of rabbits follows a sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...

which follows the pattern:
F(1) = 1
F(2) = 1
F(n) = F(n – 1) + F(n – 2)


A more visual way to see this pattern:

  

• Look at the final digit in each Fibonacci number - the units digit:
  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...
  Is there a pattern in the final digits?
  0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, ...
  Yes! In fact, the series is just 60 numbers long and then it repeats the same sequence again and again all the way through the Fibonacci series - for ever. We say the series of final digits repeats with a cycle length of 60.
  
• Suppose we look at the final two digits in the Fibonacci numbers. Do they have a pattern?
  
  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...
  Yes, there is a pattern here too. After Fib(300) the last two digits repeat the same sequence again and again. The cycle length is 300 this time. So what about the last three digits?
  and the last four digits?
  and so on??
• For the last three digits, the cycle length is 1,500
• for the last four digits,the cycle length is 15,000 and
• for the last five digits the cycle length is 150,000
• and so on...