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The Fibonacci sequence is defined as follows: $F_{0} = 0, F_{1} = 1,$ and for all integers $n \geq 2, F_{n} = F_{n−1} + F_{n−2}$. Then which of the following statements is FALSE?

  1. $F_{n+2} = 1 + \sum ^{n}_{i=0} F_{i}$ for any integer $n \geq 0$
  2. $F_{n+2} \geq \emptyset^{n}$ for any integer $n \geq 0$, where $\emptyset=\left(\sqrt{5}+1\right) / 2$ is the positive root of $x^{2} -x - 1= 0$.
  3. $F_{3n}$ is even, for every integer $n \geq 0$.
  4. $F_{4n}$ is a multiple of $3$, for every integer $n \geq 0$.
  5. $F_{5n}$ is a multiple of $4$, for every integer $n \geq 0$.
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$$\begin{array}{|c|c|c|c|c|c|c|c|} \hline \textbf{$F_0$} & \textbf{$F_1$}& \textbf{$F_2$} & \textbf{$F_3$} & \textbf{$F_4$} & \textbf{$F_5$} & \textbf{$F_6$} & \textbf{$F_7$}\\\hline \textbf{$0$} & \text{$1$}& \text{$1$} & \text{$2$} & \text{$3$} & \text{$5$} & \text{$8$} & \text{$13$}\\\hline\end{array}$$

Option (e) is FALSE.  $F_{5n}$ is a multiple of $4$, for every integer $n\ge 0.$  

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F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610
All the options are TRUE except (e).Option (e) is FALSE.

The correct answer is (e).

Answer:

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