Giving away $500$ pencils: $1$ on the first day, $2$ on the second day, etc. How long will it take?

Question

I'd like to know the name of this subject so that I discover how to solve the problem on my own.

I suppose that's something like "$(n-1) + (n-2) + \cdots$" but I'm not sure about it. =\

I have 500 pencils and I want to give them to a friend, but not at once. I intend to give one pencil on first day; two on second one; three on third; and so on.

How long will it take to give all pencils? In the end, will remain any pencils?

I appreciate the support!

Answer 1

After $n$ days, you have given $$1+2+3+\dots n=\frac{n(n+1)}2$$ We are looking for the smallest integer $n$ such that $$\frac{n(n+1)}2\geq500$$ Since $$\frac{31(31+1)}2=496 \quad\text{and}\quad\frac{32(32+1)}2=528$$ You'll give your last pencil on the 32th day, but you'll only give 4 pencils on the last Day.

Answer 2

Hint: $500 = \frac{days^2}{2}$ since however many pencils recieved on the last day, if you take half of that that will be the average you received each day over the entire time period.