Sum of Integers

Now we will try our hand at a numerical solution. Legend has it that a precocious, young Carl Frederich Gauss was restless in class so his teacher, thinking he could keep him occupied, challenged the boy to add all the numbers from 1 to 100. Before he knew it, however, the young Gauss brought his answer of 5,050 to his teacher. Astounded by the speed with which he came up with his answer, the teacher inquired and the boy explained:

He wrote each number twice, once forward, and once in reverse, then added each number together. By matching up the numbers like this, he only had to add the number 101 a total of 100 times. This is just a simple multiplication problem. Then all he had to do was divide by 2 (since every number was added twice). We can do the same thing adding the first n consecutive integers. Adding the terms in this list, together with the list in reverse, we are adding the number n + 1 a total of n times. Dividing by 2, we find the total sum.