## Problems in Mathematics and Youth Magazine, 2007 , Issue 9

For Lower Secondary Schools

1. The first $100$ positive integer numbers are written consecutively in a certain order. Call the resulting number $A$. Is $A$ a multiple of $2007$?

## Solution of problem T12/363 in M&Y

Let $f:\mathbb{N}\to\mathbb{R}$ be a function such that $f(1)=\dfrac{2007}{6}$ and $\dfrac{f(1)}{1}+\dfrac{f(2)}{2}+\cdots+\dfrac{f(n)}{n}=\dfrac{n+1}{2}\cdot f(n)\forall n\in\mathbb{N}$. Find the limit $\lim_{n\to\infty} (2008+n)f(n)$.

My solution.