Here are some patterns people have already noticed in the final digits of the Fibonacci numbers:
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!
It takes a while before it is noticeable. 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…
Patterns in the Fibonacci Numbers
