**Coded Data Proofs (4):**

Say that: y=x/k + C

And that: x={p, q} and y={p/k+C, q/k+C}

If this is the case:

And also:

Therefore, you’d have to say that:

**Coded Data Proofs (4):**

Say that: y=x/k + C

And that: x={p, q} and y={p/k+C, q/k+C}

If this is the case:

And also:

Therefore, you’d have to say that:

**Coded Data Proofs (3):**

Say y=kx+C and also that:

x={p, q} and y={kp+C, kq+C}

This would mean that:

And if the above is true:

Therefore:

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:

**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:

**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:

**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:

**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:

**Alternative method:**

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

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:**

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:

**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:**

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:

**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:**

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:

**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:**

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:

**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:**

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:

**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

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:

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