Gauss Again
 

Karl Friedrich Gauss (1777-1855) was a mathematical genius. His genius was apparent even from his early years. It was reported that he taught himself how to read and write by age 3.

An interesting story about Gauss concerns one of his achievements while in elementary school. His teacher wanted to keep the students busy while grading the students'work. The teacher assigned the students the task of finding the sum of the numbers from 1 to 100. Gauss came up with the correct sum in less than a minute. How did he do it?

For us to solve this problem, we will use the helpful problem-solving technique of working a similar but simpler problem. What is the sum of the numbers from 1 to 10? How can you align the numbers from 1 to 10 so that each set of numbers has the same sum? Multiply the sum of each set of numbers times the number of sets you created. What is the sum?

Now we will consider the numbers from 1 to 100. You can create fifty pairs of numbers where each pair would have the same sum.  What is that sum? Since there are fifty pairs, the sum of all of the numbers is fifty times that sum. What is the total sum?

To add the numbers from 1 to 100, pair up the numbers n (where n equals 1) and (101-n). If you add  n + (101-n), you get 101 for the sum. Since there are 100 numbers, there are 50 pairs. Therefore, the sum is 50(101)=5050. Remember that Gauss figured this out in elementary school in a very short time.

What is the formula for the sum of the numbers from 1 to n?  (HINT: Write 50 and 101 in terms of 100, and then change 100 to n).

How else could you have computed the sum?

Use the formula you created above to compute the sum of the numbers from 1 to 50.

Does this formula work for computing the sum of the numbers from 1 to 99? If so, compute the sum.

What is the sum of the even numbers from 2 to 200?