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JEE Main · 2024 · 09 Apr 2024 · Shift 1 · Sequences and Series

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

Answer: 1010

Solution

f (m + n) = f (m) + f (n)

with

f (1) = 1

gives

f (x) = x

. Then

\sum_{k = 1}^{2022} (λ + k) = 2022 λ + \frac{2022 \cdot 2023}{2} \leq 202 2^{2}

\Rightarrow λ \leq 2022 - 1011.5 = 1010.5

. The largest natural number is

λ = 1010

.

JEE Main · 2024 · 09 Apr 2024 · Shift 1 · Integer answer

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