(Solved): Carl Friedrich Gauss (17771855) was a mathematician who told a childhood story of how he discove ...

Carl Friedrich Gauss $(1777?1855)$ was a mathematician who told a childhood story of how he discovered a pattern and surprised his primary school teacher. His teacher had given him the task of finding the sum of the first 100 counting numbers before he could go out and play. Dr. Gauss admitted that this may have been a punitive measure for some childhood wrongdoing. Young Carl found the sum to be 5,050, but how did he do it so quickly? He looked at the problem in a different way! Instead of starting at 1 and following it with 2 , he looked at the beginning and at the end at the same time. He thought about the first and last number of the sum and added them together first. Knowing that addition is commutative, he knew he could add the 100 numbers in whatever order he pleased. $1+100=101$. He then added the 2 nd term and the 99 th term. $2+99=101$ also! Upon realizing that these numbers had a symmetry to them, could be paired up as shown in the following, and he reconceptualized what the question was. His new question was, how many pairs, which each sum to 101 , are there in this list? Since it is similar to folding the 100 -term sum in half, there are 50 of these pairs. $50(101)$ is 5,050 . Use Gauss' method to determine the sum of the first 200 counting numbers.