Age 16 to 18 Challenge Level:
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?
Can you find any numbers which are not polite?
There is actually a rather simple rule which determines whether a
given number is polite. Can you find this rule? Can you prove that
this is the case?