Sets, Relations & Functions
Cauchy Functional Equation
nta_pyq_2024_apr
Grade 11
Question:
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 ________.
Step-by-Step Solution
Key Concept: $f(x)=x$ (Cauchy equation with $f(1)=1$). $\sum_{k=1}^{2022}(\lambda+k)=2022\lambda+\frac{2022\times2023}{2}\leq2022^2$.
$\lambda\leq1010.5\Rightarrow\lambda=1010$.
Correct Answer: 1010