Chocolate Chips

     Everyone knew that Charlie had a "sweet tooth", especially for anything with chocolate in it. So no one was very surprised to hear of his recent project. The only surprise was the manner he conducted it.

     He decided to eat one chocolate chip on the first day, then 2 chips on the second day, 3 on the 3rd day. And so on. Each day he ate one more than the day before.

     By the last day of this current month, he will have consumed a total of 1035 chips.

     When did Charlie start this sweet journey?

Extra: On what date did Charlie reach the half-way point in his project? And which chip (1st, 2nd, 3rd, etc.) was it that day?

[Note: Since this is an "algebra" problem, it is expected that you will use normal algebra to solve it. This means that no brute force methods, such as spreadsheets or using a calculator to merely add up the numbers 1, 2, 3, 4, and so on, will be accepted. There is a famous formula that you should know about that automatically finds the sum of an expression like:

1 + 2 + 3 + ... + n = ?

We hope you can find and use it.]

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