I facilitated the following task with pre-service math teachers:
Take the sum of any three consecutive numbers. Do you notice anything special? Write a clear conjecture. Then write a clear proof for your conjecture.
Now, take the sum of any amount of consecutive numbers. Can you broaden your conjecture from problem 1? Prove your conjecture.
I left the task open because I wanted students to create a variety of conjectures and proofs for whole class discussion. For task 2, one student came up with the following conjecture: "The sum of $n$ consecutive integers is divisible by the greatest prime factor of $n$". I'm curious if anyone has a proof or counterexample for this claim as I do not.