If a function $$f$$ satisfies $$f(m + n) = f(m) + f(n)$$ for all $$m, n \in \mathbb{N}$$ and $$f(1) = 1$$, then the largest natural number $$\lambda$$ such that $$\sum_{k=1}^{2022} f(\lambda + k) \leq (2022)^2$$ is equal to _________

 The equation $$f(m+n) = f(m) + f(n)$$ is Cauchy’s functional equation. 

For natural numbers, this implies $$f(n) = c \cdot n$$. 

Given $$f(1) = 1$$, we find $$c = 1$$.$$f(n) = n$$.

$$\sum_{k=1}^{2022} f(\lambda + k) = \sum_{k=1}^{2022} (\lambda + k)$$

$$2022\lambda + \frac{2022(2022 + 1)}{2} = 2022\lambda + 1011(2023)$$

inequality:

$$2022\lambda + 1011(2023) \leq (2022)^2$$

$$\lambda + \frac{2023}{2} \leq 2022$$

$$\lambda + 1011.5 \leq 2022$$

$$\lambda \leq 1010.5$$

Since $$\lambda$$ must be a natural number, the largest value is 1010