Problems

Is $F_{24}$ a multiple of $3$?

We have a sequence where the first term ($x_1$) is equal to $2$, and each term is $1$ minus the reciprocal of the previous term (which we can write as $x_{n+1}=1-\frac{1}{x_n}$).

What’s $x_{57}$?

Let $n$ be a positive integer. Can $n^7-77$ ever be a Fibonacci number?

Does there exist an irreducible tiling with $1\times 2$ rectangles of a $6\times 6$ rectangle?

Irreducibly tile a floor with $1\times 2$ tiles in a room that is a $6\times 8$ rectangle.

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 $F_{n+1}$?