Tag Archives: adding

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

How to add up odd numbers from 0 upwards

In this post, I’ll be demonstrating how to add up all the odd numbers from 0 to any specific odd number. To create a robust demonstration, I’ll be taking the footsteps below:

  • I’ll first be showing you how to add up all the odd numbers from 0 to 1, using a diagram and formula.
  • I’ll then be showing you how to add up all the odd numbers from 0 to 3, using a diagram and formula.
  • I’ll also be showing you how to add up all the odd numbers from 0 to 5, using a diagram and also the same formula which was used to count up all the odd numbers from 0 to 1 and 0 to 3.
  • And finally, I’ll be using similar diagrams and formulas used to count odd numbers from 0 to 1, 0 to 3 and 0 to 5 to count odd numbers from 0 to 7 and 0 to 9.

What you will find, after I complete the tasks above – is that a pattern emerges. You will notice that the formula I use to count odd numbers from 0 to n (n which is an odd number) is very robust and will allow you to count all the odd numbers from 0 to n very easily.


COUNTING ALL THE ODD NUMBERS FROM 0 to 1:

part_1

If you count all the odd numbers from 0 to 1, what you will get is obviously 1. Furthermore, what you will also get as a formula (if n=1, H=Height and L=Length) is:

\left\{ H\cdot L \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\left\{ \left( n+1 \right) \cdot \frac { \left( n+1 \right)  }{ 2 }  \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\frac { { \left( n+1 \right)  }^{ 2 } }{ 4 } 

*If you plug the value 1 into n, you will get 1. 1 is the value of all the odd numbers added up from 0 to 1.


COUNTING ALL THE ODD NUMBERS FROM 0 to 3:

part_2

If you count all the odd numbers from 0 to 3, what you will get is 4. Furthermore, what you will also get as a formula (if n=3, H=Height and L=Length) is:

\left\{ H\cdot L \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\left\{ \left( n+1 \right) \cdot \frac { \left( n+1 \right)  }{ 2 }  \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\frac { { \left( n+1 \right)  }^{ 2 } }{ 4 } 

*If you plug the value 3 into n, you will get 4. 4 is the value of all the odd numbers added up from 0 to 3.


COUNTING ALL THE ODD NUMBERS FROM 0 to 5:

part_3

If you count all the odd numbers from 0 to 5, what you will get is 9. Furthermore, what you will also get as a formula (if n=5, H=Height and L=Length) is:

\left\{ H\cdot L \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\left\{ \left( n+1 \right) \cdot \frac { \left( n+1 \right)  }{ 2 }  \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\frac { { \left( n+1 \right)  }^{ 2 } }{ 4 } 

*If you plug the value 5 into n, you will get 9. 9 is the value of all the odd numbers added up from 0 to 5.


COUNTING ALL THE ODD NUMBERS FROM 0 to 7:

part_4

If you count all the odd numbers from 0 to 7, what you will get is 16. Furthermore, what you will also get as a formula (if n=7, H=Height and L=Length) is:

\left\{ H\cdot L \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\left\{ \left( n+1 \right) \cdot \frac { \left( n+1 \right)  }{ 2 }  \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\frac { { \left( n+1 \right)  }^{ 2 } }{ 4 } 

*If you plug the value 7 into n, you will get 16. 16 is the value of all the odd numbers added up from 0 to 7.


COUNTING ALL THE ODD NUMBERS FROM 0 to 9:

part_5

If you count all the odd numbers from 0 to 9, what you will get is 25. Furthermore, what you will also get as a formula (if n=9, H=Height and L=Length) is:

\left\{ H\cdot L \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\left\{ \left( n+1 \right) \cdot \frac { \left( n+1 \right)  }{ 2 }  \right\} \cdot \frac { 1 }{ 2 } \\ \\ =\frac { { \left( n+1 \right)  }^{ 2 } }{ 4 } 

*If you plug the value 9 into n, you will get 25. 25 is the value of all the odd numbers added up from 0 to 9.


THE FORMULA WHICH CAN BE USED TO ADD UP ALL THE ODD NUMBERS FROM 0 TO n, WHEREBY n IS AN ODD NUMBER:

If you look at each and every diagram and formula above, what you will notice is that the formula

Formula=\frac { { \left( n+1 \right) }^{ 2 } }{ 4 }

will allow you to add up all the odd numbers from 0 to n, whereby n is an odd number. The diagrams above have demonstrated why this formula is robust and completely logical. If you need to add up all the odd numbers from 0 to n (n is an odd number), the formula above is one you can trust.


ALTERNATIVE METHOD:

Using the table below, we can come up with an alternative method of calculating every odd number from 0 to n (n is an odd number):

n: Sum Total Total (Exponential form)
1 1 1 1^2
3 1+3 4 2^2
5 1+3+5 9 3^2
7 1+3+5+7 16 4^2
9 1+3+5+7+9 25 5^2

It turns out that:

image_1

*Note that 2x+1 can be used to denote an odd number.