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:part_1

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:

part_2

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:

part_3

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:

part_4

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:

image_2

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