All the tickets for the first row in a theater were sold. The seats are numbered consecutively, starting with 1. An extra ticket was sold for 1 seat by mistake. The sum of the seat numbers on all the tickets sold for that row is equal to 857. What's the number of the seat for which 2 tickets were sold? So since the seats are numbered consecutively, that means the sum of the seats has to look something like 1 plus 2 plus 3 all the way up to some n, and then plus that one extra seat that was sold by mistake. Well, we know that the sum of integers from 1 to n is equal to n times n plus 1 over 2. And when n gets large, n and n plus 1 are like pretty much the same. And so this guy is approximately equal to n squared over 2. And so now we want this n squared over 2 to be somewhere close to 857. So let's see if we can figure out a value of n that does that. Well, if I multiply 857 by 2, I get something close to 1600, which is 40 squared. So let's see what happens when we use n equals 40 and try going from there. When I plug in 40, I get 40 times 41 over 2 is 820, which is less than 857. And if you add 41 to this, it's more than 857. That means there should be exactly 40 seats, and whatever amount is left over is the extra seat that was sold. Well, we see that comparing 857 with 820, there's exactly 37 that's missing. And so that has to be the seat that we sold twice.

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