Project 6 Extension (sum of finite arithmetic series): Carl Gauss was a famous mathematician born in the late 1700s.  The story is told that when he was a young child (around third grade), his teacher posed the problem to find the sum of the first 100 integers (as some sort of punishment) but Gauss noticed a pattern and was able to solve this problem much easier than intended.  There is still debate on whether the story is true, but let’s take a look at one of our examples and investigate the pattern.

arithmeticSeries(5.0, 2.0, 6) returns 60.0

The corresponding series would be calculated as follows:

5.0 + 7.0 + 9.0 + 11.0 + 13.0 + 15.0

The pattern:

When we add the first term in the sequence with the last term, we get 20 (5.0 + 15.0).

When we add the second term in the sequence with the second to last term, we get 20 (7.0 + 13.0)

When we add the third term in the sequence with the third to last term, we get 20 (9.0 + 11.0)

There are three pairs of numbers all with the same sum of 20.  Therefore, the series would be equal to 60.

Gauss Example

1 + 2 + 3 + 4 + ….. 96 + 97 + 98 + 99 + 100

There would be 50 pairs each with a sum of 101.  Therefore, the sum of the first 100 integers is 5,050.

The formula for an arithmetic series is given above where a1 is the first number, an is the last number and n is how many numbers in the sequence.

Our example: 625.0+15.0=60

Gauss example: 10021+100=5,050