In how many ways is possible to write a number as the ordered sum of $1$ and $2$

It's a relatively simple proof. We know that their is one way to add 1 and 2 up to get to 0, namely $0=0$. So $G_0 = 0$. A similar thing holds for $G_1 = 1$. Now let's assume we know $G_{n-1}$ and $G_{n-2}$. For every combination represented in $G_{n-1}$ we can add 1 to get combinations in $G_{n}$. We can similarly add 2 to each combination in $G_{n-2}$. These combinations are distinct since they add up different numbers. Therefore $G_n = G_{n-1} + G_{n-2}$, the same recurrence relation the Fibonacci numbers have. Since $G_0 = G_1 = 1$, the same initial values of the Fibonacci sequence, we know that the sequences will be identical.