fibonacci-numbers
Given $F_m$ be the $m^\text{th}$ number in the Fibonacci sequence. Prove that for all natural $n$, $$\large |F_n^2 + F_nF_{n + 1} – F_{n + 1}^2| = 1$$
(When I'm bored, I do random stuff.)
There has been a solution below if you want to check out. And I would be appreciated if there are other solutions.
This identity is clear from the following diagram:
(imagine here a generalized picture with $F_i$ notation)
The area of the rectangle is obviously
$$F_n(F_{n}+F_{n-1})=F_nF_{n+1}$$
On the other hand, since the area of a square is x^2, it is obviously:
$$F_1^2+F_2^2+\dots+F_n^2$$
Therefore:
$$F_1^2+F_2^2+\dots+F_n^2=F_nF_{n+1}$$
You can even convert this graphical proof to an inductive proof - your inductive step would consist of adding a square $F_{n+1} * F_{n+1}$.
Best Answer
First and foremost, $$F_1^2 + F_1F_2 - F_2^2 = 1^2 - 1 \cdot 1 + 1^2 = 1$$
Assuming that the above statement is correct with $n = p \in \mathbb Z^+$. We have that $$F_{p + 1}^2 + F_{p + 1}F_{p + 2} - F_{p + 2}^2 = F_{p + 1}^2 + F_{p + 1}(F_p+ F_{p + 1}) - (F_p + F_{p + 1})^2$$
$$ = -F_p^2 - F_pF_{p + 1} + F_{p + 1}^2$$
$$ \implies |F_{p + 1}^2 + F_{p + 1}F_{p + 2} - F_{p + 2}^2| = |F_{p + 1}^2 + F_{p + 1}F_{p + 2} - F_{p + 2}^2| = 1$$
Using mathematical induction, for all natural $n$, $|F_n^2 + F_nF_{n + 1} - F_{n + 1}^2| = 1$