# How to add up all the even numbers from 0 onwards quickly

In this post, I’ll be demonstrating how you can add up all the even numbers from 0 onwards.

Adding up all the even numbers from 0 to 2:

In this diagram, we are going to say that n=2. The height of the rectangle is (n+2) and its length is n/2. This means that the area shaded in red, which is in fact equal to all the even numbers from 0 to 2 added up, is:

$\left\{ \left( n+2 \right) \cdot \frac { n }{ 2 } \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\frac { n\left( n+2 \right) }{ 4 }$

Adding up all the even numbers from 0 to 4:

In this diagram, we are going to say that n=4. The height of the rectangle is (n+2) and its length is n/2. This means that the area shaded in red, which is in fact equal to all the even numbers from 0 to 4 added up, is:

$\left\{ \left( n+2 \right) \cdot \frac { n }{ 2 } \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\frac { n\left( n+2 \right) }{ 4 }$

Adding up all the even numbers from 0 to 6:

In this diagram, we are going to say that n=6. The height of the rectangle is (n+2) and its length is n/2. This means that the area shaded in red, which is in fact equal to all the even numbers from 0 to 6 added up, is:

$\left\{ \left( n+2 \right) \cdot \frac { n }{ 2 } \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\frac { n\left( n+2 \right) }{ 4 }$

Adding up all the even numbers from 0 to 8:

In this diagram, we are going to say that n=8. The height of the rectangle is (n+2) and its length is n/2. This means that the area shaded in red, which is in fact equal to all the even numbers from 0 to 8 added up, is:

$\left\{ \left( n+2 \right) \cdot \frac { n }{ 2 } \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\frac { n\left( n+2 \right) }{ 4 }$

What we’ve discovered:

We’ve discovered that a simple formula can be used to add up all the even numbers from 0 to “n”, whereby “n” is an even number. This formula is:

$\left\{ \left( n+2 \right) \cdot \frac { n }{ 2 } \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\frac { n\left( n+2 \right) }{ 4 }$

Alternative method:

There is also an alternative formula you can use to add up even numbers, from 0 onwards. That is: