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# help

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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?

Feb 10, 2019

#1
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The 100th Lucas number = 489,526,700,523,968,661,124  mod 8 =4

Feb 11, 2019
#2
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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?

$$\begin{array}{|r|r|r|} \hline n & L_n & L_n \pmod{8} \\ \hline 1 & 1 & \color{green}1 \\ 2 & 3 & \color{green}3 \\ 3 & 4 & \color{green}4 \\ 4 & 7 & \color{red}-1 \\ 5 & 11 & \color{red}3 \\ 6 & 18 & \color{red}2 \\ 7 & 29 & \color{red}-3 \\ 8 & 47 & \color{red}-1 \\ 9 & 76 & \color{red}-4 \\ 10 & 123 & \color{red}3 \\ 11 & 199 & \color{red}-1 \\ 12 & 322 & \color{red} 2 \\ \hline 13 & 521 & \color{green}1 \\ 14 & 843 & \color{green}3 \\ 15 & 1364 & \color{green}4 \\ 16 & 2207 & \color{red}-1 \\ \ldots \\ 100 & 792070839848372253127 & -1 \text{ or } 7 \\ \hline \end{array}$$

$$\text{The cycle of L_n \pmod{8} is \mathbf{12} } : \color{green}1,\ \color{green}3,\ \color{green}4,\ \color{red}-1,\ \color{red}3,\ \color{red}2,\ \color{red}-3,\ \color{red}-1,\ \color{red}-4,\ \color{red}3,\ \color{red}-1,\ \color{red} 2$$

$$\text{To n = 100 we have \left\lfloor\dfrac{100}{8}\right\rfloor = 12 full cycles and a remainder of 4.} \\ \text{And the 4th value in the cycle is  \mathbf{-1}}$$

$$\text{The remainder when the 100th term of the sequence L_{100}=792070839848372253127 is divided by 8 }\\ \text{is \mathbf{-1} or what's the same is  \mathbf{7} }$$ Feb 11, 2019
#3
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Thanks, heureka.....I thought there might be a cyclic pattern to this.....!!!!   CPhill  Feb 11, 2019
#4
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Thanks, CPhill heureka  Feb 12, 2019