The way this site works is that if you bid, you have to pay that amount, even if you lose. Bids are incremented by 1 cent. Let’s say an item sells for 10 cents. The guy who bid 1 cent still has to pay that, the guy who bid 2 cents still has to pay that, and so forth. So, what does the auction site actually earn for selling that item for 10 cents?

Notice that 1+2+3+4+5+6+7+8+9+10 can be added up by grouping numbers from the opposite ends: \((1+10) + (2+9) + (3+8) + (4+7) + (5+6)\) This is just \(11+11+11+11+11\) or 11*5 = 55. Note that when x = 10, and we ended up multiplying \(11*5\) for the series sum.

So, the general formula is:

1 + 2 + 3 + … + n = \(\displaystyle\sum\limits_{x=0}^n x = (n+1)\frac{n}{2} = \frac{n^2+n}{2} = \frac{n(n+1)}{2}\)

Pop Quiz! If the sunglasses in the photo end up selling for $6.96, how much does the website make? \(\frac{696*697}{2}\) = \(\frac{485112}{2}\) = \(242556\) = \(\$2,425.56\) !!