What is the sum of the odd numbers from 1 to 50?

Sum of all the odd numbers from 1 to 50

625

√625 = 25

Sum of all the odd numbers from 1 to 51

Odd numbers are defined as any number that is not a multiple of 2 or has its units place digit to be odd such as 3, 45, 67, etc. The sum of odd numbers is the total summation of the odd numbers taken together for any specific range given. 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.

Sum of Odd Numbers Definition

Sum of odd numbers is defined as the summation of odd numbers taken together and added up to calculate the result. The sum of odd numbers starts from 1 and goes up to infinity. We can find the sum of odd numbers for any range such as 1 to 100, 1 to 50, and so on by using the sum of n odd numbers formula involving the concept of arithmetic progression discussed in the next section.

Sum of n Odd Numbers Formula

We know that the general form of an odd number is (2n - 1) where n 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 = n2 where n is a natural number and represents the number of terms.

Thus, to calculate the sum of 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 = n2.

Sum of First n Odd Numbers Proof

Let us now derive the sum of n odd natural numbers formula. We know that the series 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 first n odd numbers be represented as Sn = 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, last term (l) = 2n - 1 and common difference (d) = 2.

The sum of n terms of an AP is given by the formula Sn= n/2 × [a + l].

By substituting the values of 'a' and 'l' in the above formula we get,

Sn = n/2 × [1 + (2n - 1)]
Sn= n/2 × [2n]
Sn= n × n
Sn= n2

Thus, when n = 1, S1 = 12 = 1
when n = 2, S2 = 22 = 4
when n = 3, S3 = 32 = 9

Therefore, we have proved that the sum of first n odd numbers is equal to n2. 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 n2, we get, S25 = 252 = 625.

We can alternatively show this using the formula Sn = n/2 × [a + l]. We know that the sum of odd numbers 1 to 50 is represented as Sn = 1 + 3 + ... + 49. Thus, a = 1, l = 49, and n = 25.

S25 = (25/2) × [1 + 49]

= 25 × 25 = 625

Thus, the sum of odd numbers 1 to 50 is equal to 625.

Related Articles

Check these articles related to the concept of the sum of odd numbers.

• Odd Numbers
• Natural Numbers
• Arithmetic Progression
• Sum of n terms of an AP

1. 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, Sn= n/2 × [a + l]. Here, a = 1, l = 69 and n = 35 [Since there are 35 odd numbers between 1 to 70].

   ⇒ S35 = (35/2) [1 + 69]

   S35 = 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, Sn = n2.

   S35 = 352 = 1225

   Thus, the sum of odd numbers 1 to 70 is 1225.

2. 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,

   Sn = n2

   S100 = 1002

   S100 = 10000

   Thus, the sum of odd numbers from 1 to 199 is 10000.

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FAQs on 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 Formula for Sum of Odd Numbers?

We know that the series of odd numbers are always in AP as the common difference between them is 2. The formula for finding the sum of odd numbers is Sn= 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 odd numbers is Sn= n2.

How to find Sum of Odd Numbers?

The sum of odd numbers can be calculated using the formula Sn= 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 Sn= n2. To calculate the sum of odd numbers between 1 to 20 we will use Sn= n2 where n = 10 as there are 10 odd numbers between 1 to 20. Thus, S10 = 102 = 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 Sn= n2 where n represents the number of odd numbers.

How to find Sum of First n Odd Numbers?

To find the sum of first n odd numbers we can use the formula Sn= n2. For example, to calculate the sum of odd numbers between 1 to 10, we know that n = 5. Thus, S5 = 52 = 25.

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 Sn= n/2 × [a + l], where a is the first odd number and l is the last odd number. Thus, the sum of first n odd natural numbers is calculated as,
Sn = n/2 × [1 + (2n - 1)]
Sn= n/2 × [2n]
Sn= n × n
Sn= n2

What is the Mean of the Sum of n Odd Natural Numbers?

The mean of the sum of n odd natural numbers is calculated as the sum of n odd natural numbers/number of terms = n2/n = n. Thus, the mean is n.

What is the sum of an odd numbers between 1 to 50?

The odd numbers between 1 to 50 are

3, 5, 7, ......, 49

The above sequence is an A.P.

∴ a = 3, d = 5 – 3 = 2

Let the number of terms in the A.P. be n.

Then, tn = 49

Since tn = a + (n – 1)d,

49 = 3 + (n – 1)(2)

∴ 49 = 3 + 2n – 2

∴ 49 = 1 + 2n

∴ 48 = 2n

∴ n =`48/2` = 24

Now, Sn = `"n"/2 ("t"_1 + "t"_"n")`

∴ S24 = `24/2 (3 + 49)`

= 12(52)

= 624

∴ The sum of odd numbers between 1 to 50 is 624.

Concept: Sum of First n Terms of an A.P.

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