Question 109
Let $\mathbb{Z}^+$ be the set of positive integers and $(F_n)_{n \in \mathbb{Z}^+}$ be the Fibonacci sequence defined by $F_1 = F_2 = 1$ and $F_{n+1} = F_n + F_{n-1}$ for $n \ge 2$. Consider the number
$$
N = 2^{2022}F_{2023} - 2^{2021}F_{2022} + 2^{2020}F_{2021} - \dots - 2F_2 + F_1.
$$
Prove that the binary expansion of $N$ contains more 1's than 0's.
$$
N = 2^{2022}F_{2023} - 2^{2021}F_{2022} + 2^{2020}F_{2021} - \dots - 2F_2 + F_1.
$$
Prove that the binary expansion of $N$ contains more 1's than 0's.