Odd Numbers

An Odd Number is an integer that can not be divided by 2, and when divided by 2, the remainder will always be equal to 1. In other words, odd numbers refer to numbers that will always leave a remainder when divided by an even number. 

1.0Mathematical Representation of Odd numbers 

Odd numbers can mathematically be represented as 2n + 1, where n is any integer, either odd or even. Here, “n” can be positive, negative, or even zero. 

• For Example: 
• Let n = 1, then $2\times 1+1=3$
• Let n = -1, then $2\times -1+1=-1$
• Let n = 0, then $2\times 0+1=1$
• Hence, we can see that 2n +1 is a general formula of odd numbers which works for both positive and negative odd numbers. 

2.0Properties of Odd Numbers 

Addition of Odd Numbers 

• The Sum of two odd numbers will always result in an even number. For Example:
• • 3 + 5 = 8 and 27 + 5 = 32
• Similarly, the Sum of two consecutive odd numbers will also be equal to an even number, such as: 
• • 9 + 11 = 20 and 7 + 5 = 12
• The Sum of an odd number and an even number will always be equal to an odd number. For example: 
• • 8 + 9 = 17 and 5 + 4 = 9

Subtraction of Odd Numbers

• The Difference between any two odd numbers will always be equal to an even number. For example: 
• • 9 – 5 = 4 and 57 – 47 = 10
• The Difference between any two consecutive odd numbers will always be equal to 2. For Example: 
• • 9 – 7 = 2 and 11 – 9 = 2
• The difference between an even and an odd number is always equal to an odd number. For Example: 
• • 7 – 2 = 5 and 87 – 84 = 3  

Multiplication of Odd Numbers 

• The Product of any two odd numbers is always equal to an odd number. For example: 
• $3\times 7=21and5\times 5=25$
• The product of an odd number and an even number will always be equal to an even number. For Example: 
• $3\times 4=12and6\times 5=30$

Division of Odd Numbers 

• Division of an odd by two or any even number will always result in a non-integer number. For Example: 
• $\frac{5}{2}=2.5and\frac{15}{2}=7.5$

3.0List of Odd numbers 

1. 1 to 100 Odd numbers: 

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99.

2. 1 to 100 Negative Odd numbers:  

 -1, -3, -5, -7, -9, -11, -13, -15, -17, -19, -21, -23, -25, -27, -29, -31, -33, -35, -37, -39, -41, -43, -45, -47, -49, -51, -53, -55, -57, -59, -61, -63, -65, -67, -69, -71, -73, -75, -77, -79, -81, -83, -85, -87, -89, -91, -93, -95, -97, -99.

4.0Formula Related to Odd Numbers 

1. Formula to find nth terms in an Arithmetic sequence of Odd numbers: 

$n^{th}oddnumber=2n-1$

Here, n = any integer. 

2. Formula To find the terms between two given numbers a and b: 

$Numberofterms=\frac{b-a}{2}+1$

Here, 

a = Starting point of any arithmetic sequence of odd numbers. 

b = Ending point of any arithmetic sequence of odd numbers. 

3. The Sum of the “n” Odd Numbers:

The sum of the first n odd number is always equal to n².  Where n = number of odd numbers in any sequence. 

5.0Odd Vs Even Numbers 

6.0Solved Problems on Odd Numbers

Problem 1: Find the number of odd numbers from 1 to 100. 

Solution: By using the formula: 

$Numberofterms=\frac{b-a}{2}+1$

Here, a = 1 and b = 99 (as the last odd number between 1 to 100 is 99.)

$Numberofterms=\frac{99-1}{2}+1$

$Numberofterms=\frac{98}{2}+1$

$Numberofterms=49+1=50$

Problem 2: Find the sum of odd numbers from 1 to 100. 

Solution: By using the formula: 

$Sumoffirstnnumbers=n^2$

Here, n = number of odd numbers,

n = 50 

Hence, $Sumoffirstnnumbers=n^2=50^2=2500$

Problem 3: Find the number and sum of odd numbers 1 to 1000.

Solution: Here, the sequence is 1, 3, 5, 7, …., 999. 

Hence, a = 1 and b = 999

By using the formula: 

$Numberofterms=\frac{b-a}{2}+1$

$Numberofterms=\frac{999-1}{2}+1$

$Numberofterms=499+1=500$

Now, using the formula:

$Sumoffirstnnumbers=n^2$

$Sumoffirstnnumbers=500^2=250000$