What is an Odd Number?
Have you ever seen natural numbers on the number line? The representation of natural numbers on the number line is shown in the image.
If the number 2 and its multiples like 4,6,8,10 and so on are removed from the number line, what are the numbers that will be left behind on the number line? It will be 1,3,5, and so on. These numbers are called odd numbers.
We can also see that an odd number of items can not be grouped into pairs. There will always be one unpaired item left behind.
Sum of Two Odd Numbers
When two odd numbers are added the sum is always an even number.
Hence, odd + odd = even
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Formula for the Sum of Odd Numbers
The sum of odd numbers from 1 to infinity can be found easily, using Arithmetic Progression. As we know, the odd numbers are the numbers which are not divisible by 2. They are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 and so on. Now, we need to find the sum of these numbers.
Let the sum of first n odd numbers be ‘\(S_n\)’
\(S_n = 1+3+5+7+9+…………………..+(2n-1) …….\) (1)
By arithmetic progression, we know, for any series, the sum of numbers is given by,
\(S_n = \frac{1}{2} \times n[2a+(n-1)d] ……..\)(2)
Where,
\(n\) = number of digits in the series
\(a\) = First term of an A.P (arithmetic progression)
\(d\) = Common difference in an A.P
Therefore, if we put the values in equation 2 with respect to equation 1, such as;
\(a = 1, d = 2\)
Let, last term, \(l = (2n-1)\)
\(S_n = (\frac{n}{2})\times (a+l)\)
\(S_n = (\frac{n}{2})\times (1 + 2n – 1)\)
\(S_n = (\frac{n}{2})\times(2n) = n^2\)
Sum of odd numbers = \(n^2\)
Sum of 1 to 10 Odd Numbers
Number of consecutive odd numbers (n) | Sum of odd numbers (Sn) |
|---|---|
1 | \(1^2=1\) |
2 | \(2^2=4\) |
3 | \(3^2=9\) |
4 | \(4^2=16\) |
5 | \(5^2=25\) |
6 | \(6^2=36\) |
7 | \(7^2=49\) |
8 | \(8^2=64\) |
9 | \(9^2=81\) |
10 | \(10^2=100\) |
Solved Examples
Example 1: Two buckets each have an odd number of apples. Is there an even or odd number of apples in total?
Solution: We know that the sum of two odd numbers is always an even number.
Odd number + odd number = even number
Since, both buckets have an odd number of apples, hence the total number of apples will be even.
Example 2: Frank has 7 glow sticks. He bought 3 more glow sticks. How many glow sticks does Frank have?
Solution:
Frank has 7 glow sticks and he bought 3 more glow sticks.
So, total number of glow sticks = 7 + 3 = 10.
Hence, Frank has a total of 10 glow sticks.
Example 3: Find the sum of odd numbers between 1 to 20.
Solution:
The odd numbers between 1 to 20 are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.
Sum of odd numbers = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100
Hence, the sum of odd numbers between 1 to 20 is 100.
Even numbers are natural numbers that can be divided evenly, such as 2, 4, 6, 8 and so on. Even numbers are the multiples of 2, which means that even numbers can always be divided into two equal parts.
Yes, the sum of two odd numbers is always an even number.
Hence, odd number + odd number = even number
No, 6 is divisible by 2 and it is an even number.
The product of odd numbers with any even number does not result in an odd number. This is because even numbers are a multiple of 2. So the multiplication of even and odd numbers also results in multiples of 2.