If a function f satisfies f ( m + n ) = f ( m ) + f ( n ) for all m , n ? N and f ( 1 ) = 1 , then the largest natural number ? such that ? k = 1 2022 f ( ? + k ) ? ( 2022 ) 2 is equal to ___________ . [2024]

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Solution

Q.
If a function $f$ satisfies $f (m + n) = f (m) + f (n)$ for all $m,$ $n \in N$ and $f (1) = 1$ , then the largest natural number $λ$ such that $\sum_{k = 1}^{2022} f (λ + k) \leq {(2022)}^{2}$ is equal to ___________ . [2024]

Ans.
(1010)

We have, $f (m + n) = f (m) + f (n)$

So, $f (x) = k x$

$∵$ $f (1) = 1 \Rightarrow k = 1$

Hence, $f (x) = x$

Now, $\sum_{k = 1}^{2022} f (λ + k) = \sum_{k = 1}^{2022} (λ + k)$

$= \underset{2022}{\underset{⏟}{λ + λ + . . . + λ}}$ $+ (1 + 2 + . . . . + 2022)$

$= 2022 λ + \frac{2022 \times 2023}{2} \leq {(2022)}^{2}$ (Given)

$\Rightarrow λ \leq \frac{2021}{2}$

So, largest $λ = 1010$

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