Solution: Last Digit of the Sum of Fibonacci Numbers’ Squares

Solution

The table below shows the 11 Fibonacci numbers and the 11 numbers $S_n = F_0^2 +F_1^2 +\cdot \cdot \cdot+F_n^2$.

$n$	1	1	2	3	4	5	6	7	8	9	10
$F_n$	0	1	1	2	3	5	8	13	21	34	55
$S_n$	0	1	2	6	15	40	104	273	714	1870	4895

We see that $S_n = F_n \cdot F_{n+1}$. Therefore, we just find the last digit of $F_n \cdot F_{n+1}$ by applying the Pisano period method.