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Ask students to start the problem by finding the APs leading
to $544$ and $424$. Once they have represented their sums on the
number line, can they explain in words why the sums will yield
$544$ and $424$ without calculation? They can use the AP formula to
check their answers (they will need calculators).

Next students can look at the numbers $1000$ and $1001$.
Again, can they explain in words using a representation on a number
line why their answers work? They can use the AP formula to check
that they are indeed correct.

In searching for the APs, students should realise that the
factorisation of a number is important in breaking it down into an
AP. They should realise or be encouraged to relate the factors of a
number to the terms $a$ and $n$ of the arithmetical progression.
You might suggest that students also try to find APs for other
numbers such as $40, 246$ and $500$.

Finally, note that there are two possible ways of writing the
formula for the sum of an AP

$$S_n = \frac{n}{2}\left[2a+(n-1)\right] =
n\left[a+\frac{(n-1)}{2}\right]$$

In words, what are the equations saying? Which of these two
representations would be useful for different polite numbers?

In order to prove the case that a particular number is polite,
students will need to show that it can be written in one of the
above algebraic forms: essentially, they need to try to break down
each number into two appropriate factors.

Proving the final part will first require a conjecture on the part of the students. Before attempting to prove their result they might be advised to test out their conjecture on some small numbers less than $40$. This conjecturing is an important part of the mathematical process, irrespective of whether or not a final proof is constructed.

Can you explain in words why your consecutive sum will yield
the required answer? Could you see how this approach might work for
other numbers?

Why is the factorisation of the numbers important?

What is your conjecture for which numbers are polite?

Which numbers can be written as the sum of an arithmetical
progression with common difference $2$?

Some students may find it hard to find the APs leading to
$544$ and $424$ straight away, and may benefit from experimenting
with smaller numbers, perhaps using the approaches suggested in
Consecutive
Sums . This should help them to spot some patterns and make
some conjectures, which can then be investigated using the AP
formula.