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Sum of Odd Numbers
Odd numbers are defined as any number that is not a multiple of 2 or has its units place digit to be one of the digits 1, 3, 5, 7, or 9. The sum of odd numbers is the total summation of the odd numbers taken together for any specific range given.
The sum of odd numbers can be calculated easily using the sum of the arithmetic progression formula. We will be learning about the sum of odd numbers and the sum of first n odd numbers using formulas and examples in this article.
1.  What is Sum of Odd Numbers? 
2.  Sum of n Odd Numbers Formula 
3.  Sum of First n Odd Numbers Proof 
4.  Sum of Odd Numbers NOT Starting From 1 
5.  FAQs on Sum of Odd Numbers 
What is Sum of Odd Numbers?
The sum of odd numbers is defined as the summation of odd numbers taken together and added up to calculate the result. Let us examine the sum of odd numbers starting from 1 to a certain number to derive the sum of odd numbers formula.
 The sum of odd numbers from 1 to 5 = 1 + 3 + 5 (3 numbers) = 9 = 3^{2}
 The sum of odd numbers from 1 to 7 = 1 + 3 + 5 + 7 (4 numbers) = 16 = 4^{2}
 The sum of odd numbers from 1 to 9 = 1 + 3 + 5 + 7 + 9 (5 numbers) = 25 = 5^{2}
 The sum of odd numbers from 1 to 11 = 1 + 3 + 5 + 7 + 9 + 11 (6 numbers) = 36 = 6^{2}
By the above observation, we can derive a formula that "the sum of first n odd numbers is n^{2}".
We can derive this formula involving the concept of arithmetic progression discussed in the upcoming sections.
Sum of n Odd Numbers Formula
We know that the general form of an odd number is (2n  1) where n ≥ 1 is an integer. Also, consecutive odd numbers have a common difference of 2. Therefore, the series of odd numbers form an arithmetic progression. The sum of n odd numbers formula is described as follows,
Sum of n odd numbers = n^{2} (we will prove this in the next section) where n is a natural number and represents the number of terms.
Thus, to calculate the sum of the first n odd numbers together without actually adding them individually, we can use the sum of n odd numbers formula i.e., 1 + 3 + 5 +...........n terms = n^{2}. Let's derive this formula now.
Sum of First n Odd Numbers Proof
Let us now derive the sum of n odd natural numbers formula. We know that the sequence of odd numbers is given as 1, 3, 5, ... (2n  1) which forms an arithmetic progression with a common difference of 2. Let the sum of the first n odd numbers be represented as S_{n} = 1 + 3 + 5 +...+ (2n  1). Here 1 represents the first odd number and (2n  1) represents the last odd number.
Thus, the first term (a) = 1, the last term (l) = 2n  1 and the common difference (d) = 2.
The sum of n terms of an AP is given by the formula S_{n}= n/2 × [a + l].
By substituting the values of 'a' and 'l' in the above formula we get,
S_{n} = n/2 × [1 + (2n  1)]
S_{n}= n/2 × [2n]
S_{n}= n × n
S_{n}= n^{2}
Thus, when n = 1, S_{1} = 1^{2} = 1
when n = 2, S_{2} = 2^{2} = 4
when n = 3, S_{3} = 3^{2} = 9
Therefore, we have proved that the sum of first n odd numbers is equal to n^{2}. Let's take an example to understand this.
Example: Find the sum of odd numbers 1 to 50.
We know that there are 25 odd numbers between 1 to 50. Thus, by using the sum of n odd numbers formula which is n^{2}, we get, S_{25} = 25^{2} = 625.
We can alternatively show this using the formula S_{n} = n/2 × [a + l]. We know that the sum of odd numbers 1 to 50 is represented as S_{n} = 1 + 3 + ... + 49.
Thus, a = 1, l = 49, and n = 25.
S_{25} = (25/2) × [1 + 49]
= (25/2) × 50
= 25 × 25 = 625
Thus, the sum of odd numbers 1 to 50 is equal to 625.
Sum of Odd Numbers NOT Starting From 1
If we have to find the sum of odd numbers where the starting digit is NOT 1, say we have to find the sum of odd numbers from 7 to 50. If we still want to use n^{2} formula, we have to do it this way:
Required sum = (the sum of odd numbers from 1 to 50)  (the sum of odd numbers from 1 to 5)
= 25^{2}  3^{2} (∵ there are 25 odd numbers from 1 to 50 & there are 3 odd numbers from 1 to 5).
= 625  9
=616
Alternatively, it is better to use the concept of the arithmetic progression instead of n^{2} formula. For this, we list out the odd numbers from 7 to 50 which are 7, 9, ..., 49. Here 49 is the n^{th} term of AP whose first term is a = 7 and the common difference is d = 2. Now, we will first find 'n'.
a_{n} = a + (n  1) d
49 = 7 + (n  1) 2
49 = 7 + 2n  2
2n = 44
n = 22
Now, the sum of 22 terms of AP is:
S_{n} = n/2 (a + l)
= 22/2 (7 + 49)
=616
Note that we got the same answer using both methods.
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Sum of Odd Numbers Examples

Example 1: Find the sum of odd numbers 1 to 70.
Solution:
To find the sum, we can use the sum of n odd numbers formula, S_{n}= n/2 × [a + l]. Here, a = 1, l = 69 and n = 35 [Since there are 35 odd numbers between 1 to 70].
⇒ S_{35} = (35/2) [1 + 69]
S_{35} = 35 × 35 = 1225
Alternate Method:
Since there are 35 odd numbers between 1 to 70,
Thus, n = 35
According to the sum of n odd numbers formula, S_{n} = n^{2}.
S_{35} = 35^{2} = 1225
Thus, the sum of odd numbers 1 to 70 is 1225.

Example 2: Find the sum of odd numbers from 1 to 199.
Solution:
We know that, from 1 to 199, there are 100 odd numbers. Thus, n = 100. Using the formula of the sum of first n odd numbers,
S_{n} = n^{2}
S_{100} = 100^{2}
S_{100} = 10000
Thus, the sum of odd numbers from 1 to 199 is 10000.

Example 3: Find the sum of odd numbers between 0 and 50.
Solution:
We know that there are 25 odd numbers between 0 and 50.
So by sum of odd numbers formula:
the sum of odd numbers between 0 and 50 is n^{2} = 25^{2} = 625.
FAQs on Sum of Odd Numbers
What is the Meaning of Sum of Odd Numbers?
The sum of odd numbers is defined as the addition or summation of all the odd numbers present in a given range. For example, to calculate the sum of odd numbers between 1 to 10 we will consider all the odd numbers in this range and add them. 1 + 3 + 5 + 7 + 9 = 25.
What is the Sum of Odd Numbers Formula?
We know that the series of odd numbers are always in AP as the common difference between them is 2.
 The sum of odd numbers formula is S_{n}= n/2 × [a + l] where 'a' is the first odd number, 'l' is the last odd number and 'n' is the number of odd numbers present in that range.
 Another formula to calculate the sum of first n odd numbers is S_{n}= n^{2}. But to use this formula, the starting odd number should always be 1.
How to find Sum of Odd Numbers?
The sum of odd numbers can be calculated using the formula S_{n}= n/2 × [a + l] where 'a' is the first odd number, 'l' is the last odd number and 'n' is the number of odd numbers or S_{n}= n^{2}. To calculate the sum of odd numbers between 1 to 20 we will use S_{n}= n^{2} where n = 10 as there are 10 odd numbers between 1 to 20. Thus, S_{10} = 10^{2} = 100.
What is the Formula for the Sum of the First n Odd Numbers?
The formula for the sum of the first n odd numbers is given as S_{n}= n^{2} where n represents the number of odd numbers. i.e., n^{2} is the sum of odd numbers from 1 to (2n  1).
How to Find Sum of First n Odd Numbers?
To find the sum of first n odd numbers we can use the formula S_{n}= n^{2}. For example, to calculate the sum of odd numbers between 1 to 10, we know that n = 5. Thus, S_{5} = 5^{2} = 25.
What is the Sum of Odd Numbers From 1 to 100?
Note that there are 50 odd numbers from 1 to 100. i.e., n = 50. Therefore, the sum of odd numbers from 1 to 100 = n^{2} = 50^{2} = 2500.
What is the Sum of First n Odd Natural Numbers?
The sum of first n odd natural numbers can be represented as 1 + 3 + 5 + ... + (2n  1) where 1 is the first odd number and (2n  1) represents the last odd number. There are n natural numbers in this AP series. The sum of n terms of an AP is given by the formula S_{n}= n/2 × [a + l], where a is the first odd number and l is the last odd number. Thus, the sum of the first n odd natural numbers is calculated as,
S_{n} = n/2 × [1 + (2n  1)]
S_{n}= n/2 × [2n]
S_{n}= n × n
S_{n}= n^{2}
What is the Mean of the Sum of n Odd Numbers?
The mean of the sum of n odd numbers is calculated as the sum of n odd numbers/number of terms = n^{2}/n = n. Thus, the mean is n.
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