Problems

The figure shows the scheme of a go-karting route. The start and finish are at point $A$, and the driver can go along the route as many times as he wants by going to point $A$ and then back onto the circle.

It takes Fred one minute to get from $A$ to $B$ or from $B$ to $A$. It also takes one minute for Fred to go around the ring and he can travel along the ring in an anti-clockwise direction (the arrows in the image indicate the possible direction of movement). Fred does not turn back halfway along the route nor does not stop. He is allowed to be on the track for 10 minutes. Find the number of possible different routes (i.e. sequences of possible routes).

A coin is thrown 10 times. Find the probability that it never lands on two heads in a row.

Simplify $F_0-F_1+F_2-F_3+...-F_{2n-1}+F_{2n}$, where $n$ is a positive integer.

Prove that every pair of consecutive Fibonacci numbers are coprime. That is, they share no common factors other than 1.

Calculate the following: $F_1^2-F_0{F}_2$, $F_2^2-F_1{F}_3$, $F_3^2-F_2{F}_4$, $F_4^2-F_3{F}_5$ and $F_5^2-F_4{F}_6$. What do you notice?

Work out $F_3^2-F_0{F}_6$, $F_4^2-F_1{F}_7$, $F_5^2-F_2{F}_8$ and $F_6^2-F_3{F}_9$. What pattern do you spot?

Can every whole number be written as the sum of two Fibonacci numbers? If yes, then prove it. If not, then give an example of a number that can’t be. The two Fibonacci numbers don’t have to be different.

What’s $\sum _{i=0}^n{F}_i^2=F_0^2+F_1^2+F_2^2+...+F_{n-1}^2+F_n^2$ in terms of just $F_n$ and $F_{n+1}$?

What are the ratios $\frac{F_2}{F_1}$, $\frac{F_3}{F_2}$, and so on until $\frac{F_7}{F_6}$? What do you notice about them?

$\phi =\frac{1+\sqrt{5}}{2}$ is the golden ratio. Using the fact that ${\phi }^2=\phi +1$, can you express ${\phi }^3$ in the form $a\phi +b$, where $a$ and $b$ are positive integers?