TIFR CSE 2014 | Part A | Question: 3

Question

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?

• A. $F_{n+2} = 1 + \sum ^{n}_{i=0} F_{i}$ for any integer $n \geq 0$
• B. $F_{n+2} \geq \emptyset^{n}$ for any integer $n \geq 0$, where $\emptyset=\left(\sqrt{5}+1\right) / 2$ is the positive root of $x^{2} -x - 1= 0$.
• C. $F_{3n}$ is even, for every integer $n \geq 0$.
• D. $F_{4n}$ is a multiple of $3$, for every integer $n \geq 0$.
• E. $F_{5n}$ is a multiple of $4$, for every integer $n \geq 0$.

Comments

I'm surprise how this question can be categorize with tag 'easy'.The person who categorize it with this tag must try to prove option b.

useful references.

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html

https://en.wikipedia.org/wiki/Golden_ratio

---

@Rupendra

Proving may be difficult but its mcq, one can get an answer by hit and trail.

---

@smsubham

But still shouldn't be tagged as easy.

---

Can any one please elaborate on option b.

Answer 1

$$\begin{array}{|c|c|c|c|c|c|c|c|} \hline \textbf{$F_0$} & \textbf{$F_1$}& \textbf{$F_2$} & \textbf{$F_3$} & \textbf{$F_4$} & \textbf{$F_5$} & \textbf{$F_6$} & \textbf{$F_7$}\\\hline \textbf{$0$} & \text{$1$}& \text{$1$} & \text{$2$} & \text{$3$} & \text{$5$} & \text{$8$} & \text{$13$}\\\hline\end{array}$$

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

Comments

In option b, how F2 >= 1.61 at n = 0?

---

because at n=0, Φ$^{n}$=1 and F$_{2}$=1.

Answer 2

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).