# Sum of naturals (Gaussian Sum)

**1 to 100**.

We could do it by brute force, but that seems tedious and impractical.

**Fortunately**, there is a formula to sum them for us.

```
(n * ( n + 1 )) / 2
```

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(100 * 101)/2 = 5050

Simple enough, but why does it work?Let's see if we can get an intuition of why it works.

### Why Does It Work

We can think of this formula as calculating the area of a rectangle, and then halving it.
The **width** of the rectangle is **n+1** the **height** is **n** .

Why is it **n * (n + 1)** instead of **n * n**?

We need to add **+1** to **shift** our second triangle over so that it'll be the same area as the first, if we didn't add the **+1** our second triangle would start at **0 instead of 1** and end at **n - 1** instead of **n **(look at the interactive above to see it).

Now, we have constructed a rectangle with two triangles of **equal** area.

Each triangle has an area that is our answer because they are constructed by drawing **1...n.**

Therefore, we get our answer by dividing by 2.

### The Story Behind This Formula

One day a teacher gave his class some busy work, he made his students sum the numbers 1 to 100.

He was surprised when one of his students finished quickly.

The student's name was Carl Friedrich Gauss. He had devised the formula **(n * (n + 1))/2** by noticing that if you listed **1...100** then underneath it listed **100...1** each column would add up to **101**, that is **100 101 times** or **100 * 101**.

Therefore , **100 * 101 divided by 2** is the answer.

Gauss was 9 at the time.

### Resources