GAUSSIAN SUMMATION

Gaussian summation is a mathematical formula that allows you to calculate the sum of the first n positive integers quickly and efficiently. The formula is named after the German mathematician Carl Friedrich Gauss, who discovered it when he was a child.

The Gaussian summation formula

The Gaussian summation formula is:

1 + 2 + 3 + ... + n = n(n+1)/2

where n is a positive integer.

Formula demonstration

The proof of the Gaussian summation formula is simple and elegant. It can be proven as follows:

1. Write the sum of the first n positive integers in ascending order:

1 + 2 + 3 + ... + n

1. Write the same sum in descending order:

n + (n-1) + (n-2) + ... + 1

1. Add the two previous sums:

(1 + n) + (2 + n-1) + (3 + n-2) + ... + (n + 1)

1. Simplify the previous sum:

n(1 + n) = n(n+1)

1. Divide the previous sum by 2:

n(n+1)/2

Examples of Gaussian summation

The Gaussian summation formula is very useful for calculating the sum of the first n positive integers. Here are some examples:

- The sum of the first 5 positive integers is:

1 + 2 + 3 + 4 + 5 = 5(5+1)/2 = 15

- The sum of the first 10 positive integers is:

1 + 2 + 3 + ... + 10 = 10(10+1)/2 = 55

Applications of Gaussian summation

The Gaussian summation formula has many applications in mathematics and real life. Some of the applications include:

- Calculation of the sum of the first n positive integers
- Calculation of the arithmetic mean of a set of numbers
- Calculation of the sum of the squares of the first n positive integers
- Applications in statistics and probability