Tag Archives: odd numbers

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.

Multiplying Even and Odd Numbers

Today I’m going to be showing you what would happen if you were to multiply:

a) An even number by an even number;

b) An odd number by an odd number;

c) An even number by an odd number.

Firstly, let us define what an even number is:

An even number can be described using the expression 2n, whereby (n) would be a whole number ranging from 0 upwards.

Next, let us define what an odd number is:

An odd number can be described using the expression 2n+1, and similarly (as is the case with even numbers), (n) would be a whole number ranging from 0 upwards.

Now, since we’ve defined how both even numbers and odd numbers can be described in terms of mathematical expressions, let’s focus our attention on multiplying even numbers by even numbers, odd numbers by odd numbers and even numbers by odd numbers…

Multiplying even numbers by even numbers:

Let’s produce two whole numbers which could be equal to one another or not equal to one another… Let’s call these numbers { n }_{ 1 } and { n }_{ 2 }.

Using these two whole numbers we can multiply two unknown even numbers by each other in such a manner:

2{ n }_{ 1 }\cdot 2{ n }_{ 2 }

This would invariably give us the result 4{ n }_{ 1 }{ n }_{ 2 }=2\cdot 2{ n }_{ 1 }{ n }_{ 2 }. Now, the product of { n }_{ 1 }{ \cdot n }_{ 2 } would be a whole number and since this is the case, you would have to say that an even number multiplied by an even number would produce an even number. Let’s not forget that even numbers are multiples of 2.

Multiplying odd numbers by odd numbers:

Once again, let’s come up with two whole numbers which could be equal to one another or not equal to one another… These whole numbers will be { n }_{ 3 } and { n }_{ 4 }.

This would mean that two odd numbers being multiplied by one another would produce an expression as such:

\left( 2{ n }_{ 3 }+1 \right) \left( 2{ n }_{ 4 }+1 \right)

And if we expand the expression above, we’ll get:

4{ n }_{ 3 }{ n }_{ 4 }+2{ n }_{ 3 }+2{ n }_{ 4 }+1

Now if we re-arrange the expression above, we can get:

2\left( 2{ n }_{ 3 }{ n }_{ 4 }+{ n }_{ 3 }+{ n }_{ 4 } \right) +1

Since the expression 2{ n }_{ 3 }{ n }_{ 4 }+{ n }_{ 3 }+{ n }_{ 4 } must be a whole number, you would be forced to conclude that an odd number multiplied by an odd number would produce an odd number.

Multiplying even numbers by odd numbers:

We will for the last time come up with two whole numbers { n }_{ 5 } and { n }_{ 6 }.

An even number and an odd number being multiplied by one another could be shown using the mathematical expression below:

2{ n }_{ 5 }\cdot \left( 2{ n }_{ 6 }+1 \right)

Lazily, we could conclude that an even number multiplied by an odd number would produce an even number. This is because even numbers are all multiples of 2.

Ok… So, let’s summarise what we’ve discovered:

i) An even number multiplied by an even number would produce an even number;

ii) An odd number multiplied by an odd number would produce an odd number;

iii) An even number multiplied by an odd number would produce an even number.

Knowing this we can further strengthen our mathematical reasoning. 🙂