\section*{Problem 6 - 031116}

The sum of 100 consecutive natural numbers is 1000050.

What is the smallest and the largest of these numbers?
Let \(a\) be the smallest of the 100 numbers, then \(a+99\) is the largest of the numbers. The sum of the 100 numbers is

\(a+(a+1)+\ldots+(a+98)+(a+99)=(a+(a+99))+((a+1)+(a+98))+\ldots+((a+49)+(a+50))=\)

\[
=\underbrace{(2 a+99)+\ldots+(2 a+99)}_{50 \text { terms }}=100 a+4950
\]

i.e., \(100 a=995100\) or \(a=9951\) and \(a+99=10050\). Therefore, 9951 is the smallest and 10050 is the largest of the 100 numbers.