Homework 11, due Monday, September 26, 2011

Carl F. Gauss was one of the top mathematicians ever. As an elementary school child he figured out a method for adding up a sequence of numbers. For example, here is how he figured out that 1 + 2 + ... + 100 = 5050.

He wrote the original sum, then rewrote it again backward just below, and added up each column:

  1  +  2  +  3 + ...  + 99 + 100
100  + 99  + 98 + ...  +  2 +   1
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101 + 101 + 101 + ... + 101 + 101 = ?

He then figured out what the bottom row added up to (i.e., he figured out what the "?" was) and used that to find the original sum, which turns out to be 5050.

Problem [1]: Explain how to find what "?" is, and once you know that, how to get the final answer, 5050.

Problem [2]: Show how to use Gauss's method to give an explicit formula for the sum 1 + 2 + 3 + ... + n, where n is the number of terms in the sum. (So when n = 2, the sum is 1 + 2, but when n = 5 the sum is 1 + 2 + 3 + 4 + 5, etc. Your formula will be in terms of n. When you plug 2 in for n in your formula, you should get 3 since 1 + 2 = 3. When you plug 6 in for n in your formula you should get 21, since 1 + 2 + 3 + 4 + 5 + 6 = 21, etc.)