Sum of odd Numbers

Odd numbers are those that cannot be evenly divided by two. It is impossible to split it equally into two independent integers. When we divide an odd integer by two, we get a residue. Odd numbers include 1, 3, 5, 7, and so on. Even numbers are the polar opposite of odd numbers. 

Sum of odd numbers:

The sum of odd numbers is defined as the sum of odd numbers added together to get the result. The sum of odd numbers begins at 1 and continues indefinitely. We may use the sum of n odd numbers formula, which involves the concept of arithmetic progression, to compute the sum of odd numbers for any range, such as 1 to 100, 1 to 50, and so on. 

Sum of n odd numbers:

We already know that an odd number has the general form (2n – 1), where n is an integer. In addition, the difference between two successive odd numbers is two. As a result, an arithmetic progression is formed by the odd integers. Here’s how to calculate the sum of n odd numbers:

Where n is a natural integer that denotes the number of terms, the sum of n odd numbers equals n². 

Sum of ‘n’ odd numbers = Sₙ 

Sₙ = 1 + 3 + 5 +……..+ (2n-1) 

Sₙ = n² where, n = number of terms 

We can utilise the sum of n odd numbers formula, which is 1 + 3 + 5 +………..n terms = n², to determine the sum of the first n odd numbers without actually adding them separately. 

Sum of first n odd numbers:

Let’s look at the formula for the sum of n odd natural integers. The sequence of odd numbers is written as 1, 3, 5,…(2n – 1), forming an arithmetic progression with a common difference of 2. Let Sn = 1 + 3 + 5 +…+ be the sum of the first n odd numbers (2n – 1). The number 1 is the first odd number, while the number (2n – 1) is the last odd number.

As a result, the first term (a) equals 1, the last term (l) equals 2n – 1, and the common difference (d) equals 2.

The formula Sn= n/2 [a + l] gives the sum of an AP’s n terms.

By changing the values of ‘a’ and ‘l’ in the formula above, we get 

Sₙ = n/2 x [1 + (2n-1)] 

Sₙ = n/2 x [2n] 

Sₙ = n x n 

Sₙ = n² 

Therefore, when n = 1, S₁ = 1² = 1

when n = 2, S₂ = 2² = 4 

when n = 3, S₃ = 3² = 9 

As a result, we’ve established that the sum of the first n odd numbers equals n². 

Sum of even numbers:

The sum of even integers from 2 to infinity can be easily calculated using both Arithmetic Progression and the sum of all natural numbers formula. We already know that even numbers are those that are totally divisible by two. 2, 4, 6, 8, 10, 12, 14, 16, and so on are the numbers. Now we must calculate the sum of these figures. 

Sum of even numbers formula:

The sum of even numbers is calculated using the formula n(n+1), where n is the natural number. This formula can be found using the sum of natural numbers formula, such as: 

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + ……. + n

S = n(n+1)/2

We need to multiply the above formula by 2 to get the sum of successive even numbers. Hence, 

Sₙ = n(n+1) 

Sum of even numbers formula using AP:

Let Sₙ be the sum of the first n even numbers. 

Sₙ = 2 + 4 + 6 + 8 + 10 +……..+ (2n)………(1) 

We know that the total of numbers for any sequence is provided by Arithmetic Progression; 

Sn = ½ x n [2a + (n-1)d]…….(2)

Where, n = number of digits in series 

a = first term of AP

d = common difference in AP

As a result, if we put the numbers in equation 2 in relation to equation 1, we get: 

a = 2, d = 2

Let last term I = (2n)

So, we will have the sum as 

Sn = ½ n [2.2 + (n-1)2]

Sn = n/2 [4+2n-2]

Sn = n/2 [2+2n]

Sn = n(n+1) 

Therefore, sum of n even numbers = n(n+1).

Conclusion:

An even number is one that is equally divisible by two, while an odd number is one that is not evenly divided by two. When we divide a number by two and there remains some leftover that is not divisible by two again, we call it an odd number.