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/Solution: Last Digit of the Sum of Fibonacci Numbers
Solution: Last Digit of the Sum of Fibonacci Numbers
Look at the solutions for the Last Digit of the Sum of Fibonacci Numbers Problem.
We'll cover the following...
We'll cover the following...
Solution 1: Pisano period
The table below shows the first 11 Fibonacci numbers and the first 11 numbers .
Stop and think: Do you see any similarities between sequences and ?
It looks like . Let’s prove it by induction. This condition certainly holds for the base step since . For the induction step, let’s assume that the statement holds for and prove it for :
Another way of arriving at the formula is to sum up the following equalities:
Since the identically colored terms cancel out, the sum of all terms on the left is ...