JEE Mains · Maths · STD 11 - 8. sequence and series
If a function \(f\) satisfies \(f(m+n)=f(m)+f(n)\) for all \(\mathrm{m}, \mathrm{n} \in \mathrm{N}\) and \(\mathrm{f}(1)=1\), then the largest natural number \(\lambda\) such that \(\sum_{\mathrm{k}=1}^{2022} \mathrm{f}(\lambda+\mathrm{k}) \leq(2022)^2\) is equal to ..........
- A \(1010\)
- B \(1015\)
- C \(1678\)
- D \(1345\)
Answer & Solution
Correct Answer
(A) \(1010\)
Step-by-step Solution
Detailed explanation
\( \mathrm{f}(\mathrm{m}+\mathrm{n})=\mathrm{f}(\mathrm{m})+\mathrm{f}(\mathrm{n}) \) \( \Rightarrow \mathrm{f}(\mathrm{x})=\mathrm{kx} \) \( \Rightarrow \mathrm{f}(1)=1 \) \( \Rightarrow \mathrm{k}=1 \) \( \mathrm{f}(\mathrm{x})=\mathrm{x}\) Now…
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