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 λ such that Σk=1²⁰²² f(λ+k) ≤(2022)² is equal to _________.

Solution

Maths sir · 4 steps

Additivity with f(1)=1 forces f(n)=n, and after that the sum is just an arithmetic series.

1. Find f. Put n=1: f(m+1)=f(m)+1. Each step up adds 1, and f(1)=1, so f(n)=n for every natural number n.
2. Write the sum. Now f(λ+k)=λ+k. Adding over k=1 to 2022 gives 2022λ+(2022× 2023)/2.

   2022λ+(2022× 2023)/2≤ 2022²

3. Divide by 2022. This gives λ+2023/2≤ 2022, so λ≤ 2022-1011.5=1010.5.
4. So. λ is a natural number no larger than 1010.5, so round down, not up: λ=1011 would break the inequality. The largest value is 1010.

The idea to keep. On the naturals, f(m+n)=f(m)+f(n) gives f(n)=n f(1). Find f first, then the question becomes a sum of consecutive integers.

Answer from the NTA final answer key. Solution by PrepWiser.