Resource recommendation to help with deriving mathematical formulas

Hi everyone,
I am a newbie to software development and lately I've been trying to improve on my problem solving skills and one common theme that stood out to me was my inability to derive mathematical formulas that will help with my programming logic. I am not sure if this makes sense, but let me give you a few examples, and hoping to get some feedback on how to improve on this skill?

Problem 1: Multiples of 3 and 5

If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
Find the sum of all the multiples of 3 or 5 below 1000.

I wrote the following function to solve it, which still have lots of room for improvement and need to make it DRY, but I was hoping I could come up with a mathematical formula to solve it instead of using brute force.

function sum(number, max) {
let i = 1;
let result = 0;
let sum = 0;

  while (result < max) {

        result = number * i++;

    if(result < max) {
        sum = sum + result;
    }
  }

  return sum;
}   


console.log(sum(3, 1000) + sum(5, 1000));

Problem 2: Calculate sum up to n

Write a function that calculates sum up to n

I didn't write this, but someone wrote a formula for it involving n, and I have no idea how he was able to derive this.

function sum(n) {
  return n * (n+1)/2;
}

which does the same thing as this:

function sum(n) {
let total = 0;
for (let i=0; i <= n; i++) {
   total += i;
}

return total;
}

Hopefully this clearly explains my question and any recommendation would be greatly appreciated!
Thank you,