A polite number is a number which can be written as the sum of two or more consecutive positive integers. Find the two consecutive sums which produce the polite numbers $544$ and $424$.
How would you represent these sums using a number line? Use this visualisation approach to decide which consecutive sums would give rise to the polite numbers $1000$ and $1001$. Do these numbers arise as more than one consecutive sum? How do these numbers relate to the formula for the sum of an arithmetical progression?