The Lucas sequence is the sequence 1, 3, 4, 7, 11, where the first term is 1,
the second term is 3 and each term after that is the sum of the previous two terms.
What is the remainder when the 100th term of the sequence is divided by 8?
\(\begin{array}{|r|r|r|} \hline n & L_n & L_n \pmod{8} \\ \hline 1 & 1 & \color{green}1 \\ 2 & 3 & \color{green}3 \\ 3 & 4 & \color{green}4 \\ 4 & 7 & \color{red}-1 \\ 5 & 11 & \color{red}3 \\ 6 & 18 & \color{red}2 \\ 7 & 29 & \color{red}-3 \\ 8 & 47 & \color{red}-1 \\ 9 & 76 & \color{red}-4 \\ 10 & 123 & \color{red}3 \\ 11 & 199 & \color{red}-1 \\ 12 & 322 & \color{red} 2 \\ \hline 13 & 521 & \color{green}1 \\ 14 & 843 & \color{green}3 \\ 15 & 1364 & \color{green}4 \\ 16 & 2207 & \color{red}-1 \\ \ldots \\ 100 & 792070839848372253127 & -1 \text{ or } 7 \\ \hline \end{array} \)
\(\text{The cycle of $L_n \pmod{8}$ is $\mathbf{12}$ } : \color{green}1,\ \color{green}3,\ \color{green}4,\ \color{red}-1,\ \color{red}3,\ \color{red}2,\ \color{red}-3,\ \color{red}-1,\ \color{red}-4,\ \color{red}3,\ \color{red}-1,\ \color{red} 2 \)
\(\text{To $n = 100$ we have $\left\lfloor\dfrac{100}{8}\right\rfloor = 12$ full cycles and a remainder of $4$.} \\ \text{And the $4th$ value in the cycle is $ \mathbf{-1}$}\)
\(\text{The remainder when the $100th$ term of the sequence $L_{100}=792070839848372253127$ is divided by $8$ }\\ \text{is $\mathbf{-1}$ or what's the same is $ \mathbf{7}$ }\)