Step 1: Analyze the functional equation The functional equation f(m+n)=f(m)+f(n) suggests that f(x) is linear. Assume:
f(x)=kx.
Substitute f(1)=1:
f(1)=k⋅1⟹k=1.
Thus:
f(x)=x.
Step 2: Expand the summation We are given:
k=1∑2022f(λ+k)≤(2022)2.
Substitute f(x)=x:
k=1∑2022(λ+k)≤(2022)2.
Split the summation:
k=1∑2022(λ+k)=k=1∑2022λ+k=1∑2022k.
Step 2.1: Simplify each term
- k=1∑2022λ=2022⋅λ.
- k=1∑2022k=22022⋅2023(sum of the first 2022 natural numbers).
Thus:
k=1∑2022(λ+k)=2022⋅λ+22022⋅2023.
Step 3: Solve the inequality Substitute into the inequality:
2022λ+22022⋅2023≤(2022)2.
Simplify:
2022λ≤(2022)2−22022⋅2023.
Factor 2022 out:
2022λ≤2022(2022−22023).
Simplify further:
λ≤2022−22023.
Calculate:
λ≤2022−1011.5=1010.5.
Step 4: Largest natural number Since λ must be a natural number:
λ=1010.
Final Answer:- 1010.