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You've been good!

You have been very good and your parents have promised to reward you by giving
you one candy bar on the first of ten days in a row that you continue to be
good, two on the second, three on the third and so on for ten days in a row.
On the tenth day of consistently good behavior you will get ten bars. Assuming
that you are good ten days in a row, how many candy bars do you expect to collect
in total?

There are several ways to solve this problem. We could add
        1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
to find the total, or we could draw a picture and count up the bars. The picture
below shows ten days and the number of bars you get each day. Count them up and
you will find the number of bars you will get in total.

      Days You Are Good
    1 2 3 4 5 6 7 8 9 10
    0 0 0 0 0 0 0 0 0 0
      0 0 0 0 0 0 0 0 0
        0 0 0 0 0 0 0 0
          0 0 0 0 0 0 0
            0 0 0 0 0 0
              0 0 0 0 0
                0 0 0 0
                  0 0 0
                    0 0
                      0

How many did you count? Did you get 55? If you didn't you better count again,
because that is the correct answer.

There is a third way to solve this problem. Instead of adding the numbers 1
through 10 in order, we could add them in pairs starting at the ends and moving
toward the center. Lets see what happens when we do this. The result may surprise
you.

        1 + 10 = 11
        2 +  9 = 11
        3 +  8 = 11
        4 +  7 = 11
        5 +  6 = 11

Now if we add those 5 elevens together we get 11+11+11+11+11 = 55. This is the
same number as before. Cool!