Gimmicky Bidding Websites (Series/Sequence)

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?  $\frac{696*697}{2}$ = $\frac{485112}{2}$ = $242556$ = $\2,425.56$   !!