Sum of Odd numbers

The sum of odd numbers is adding the consecutive odd numbers together. The sum of odd numbers formula is s_n = n². By using the formula, we can easily calculate the sum of odd numbers from 1 to infinity. For instance, the sum of the first five consecutive odd numbers is: 1 + 3 + 5 + 7 + 9 = 25.

Odd numbers are the numbers that are not divisible by 2; the sum of n odd numbers can be calculated using the formula:

The sum of n odd numbers = n², where n is the number of odd numbers

For example, let’s find the sum of the first 5 odd numbers

The sum of n odd numbers = n²

Here, n = 5

So, the sum of the first five odd numbers = 5² = 25

The sum of the first n odd numbers formula is n². To understand how we get this formula, let’ look at the pattern of odd numbers. The general form of odd numbers is 2n -1 where the common difference is 2. 

For the sequence of odd numbers: 1, 3, 5, 7,… (2n - 1)

d = 2

So, s_n = 1 + 3 + 5 + 7 + …. + (2n - 1)

The sum of n terms of an arithmetic sequence is: S_n = n/2 (2a + (n - 1)d)

Substituting the value of a and d, here, a = 1, d = 2(3 - 1 = 2)

S_n = n/2 (2 × 1 + n - 1)2)

= n/2 (2 + 2n - 2)

= n/2 × 2n

= n²
 
So, the sum of n odd numbers is n².

Now we will find the sum of odd numbers not starting from 1. So let’s find the sum of odd numbers from 11 to 60.
 

The sum of the first n odd numbers is: S_n = n²

The sum of odd numbers not starting from 1, S_n = (n/2)(a + l), where a is the first term and l is the last term. 

Here, we will find the sum of n odd numbers from 11 to 60, so the sequence is 11, 13, 15, 17, 19,…, 59

To find the sum, first we will find n. Here, a = 11 and d = 2

a_n = a + (n - 1)d

59 = 11 + (n - 1)2

59 = 11 + (2n - 2)

59 = 2n + 9

2n = 59 - 9

2n = 50

n = 50/2 

= 25

Here, n = 25

So, s_n = n/2 (a + l)

Where l is the last term 

= 25/2(11 + 59) 

= 25/2 + (70) 

= 25 × 35 

= 875

The natural numbers are the counting numbers starting from 1. The sum of n natural numbers is: S_n = n(n + 1)/2

Let’s find the sum of the first 10 natural numbers.

Here, n = 1 and d = 1

So, s_n =   n(n + 1)/2

= 10(10+ 1)/2 = 10 × 11/2

=110/2 = 55

Even numbers are the numbers that are evenly divisible by 2. The sum of the first n even numbers (2, 4, 6, 8,…, 2n)can be calculated using the formula: S_n = n(n + 1), where n is the number of terms. 

Let’s find the sum of the first 10 even numbers.

Here, n = 10

So, S₁₀ = 10(10 + 1) 

=10 × 11 

= 110

The sum of the squares of n natural numbers can be calculated using the formula: (n(n + 1)(2n + 1))/6. Where n is the number of terms. 

For example, let’s find the sum of the squares of 5 natural numbers

The first 5 natural numbers are: 1, 2, 3, 4, 5

The sum of squares of n natural numbers: S₅ = (5 (5 + 1) ((2×5) + 1))/6

= (5 (6) (11))/6

= 55

The GP can be represented as a, ar, ar², …, ar^{n - 1}, where a is the first term, r is the common ratio. The sum of n terms of GP: S_n = a(1 - rⁿ)/(1 - r), where r ≠ 1. 

Sum of infinite terms of GP: S_n = a/(1 - r), where |r| < 1.

In AP, the difference between any two consecutive terms will be the same. The sum of n terms of an AP is: S_n = (n/2)(2a + (n - 1)d), where ‘a’ is the first term and ‘d’ is the common difference.    

By mastering the sum of odd numbers, students can make calculations faster and improve their mental math skills. Here are some tips and tricks to master the sum of odd numbers. 

• Memorize the formula to make the calculation easier: S_n = n².

• Identify the pattern: The sum of a few odd numbers is: 1, 3, 5, 7, 9, …, and their sum form perfect squares:
  1 = 1² 
  1 + 3 = 4 = 2²
  1 + 3 + 5 = 9 = 3²

• You can also use the sum of arithmetic progression formula S_n = (n/2)(2a + (n - 1)d) to find the sum of odd numbers in a sequence. 

• Odd numbers: Numbers that are not divisible by 2 are considered odd numbers. For example, 1, 3, 5, 7, .. 

• Sum: The result of adding two or more numbers is the sum

• Arithmetic progression: An arithmetic progression is a sequence of numbers where the difference between any two consecutive numbers is always the same.

• Common difference: The difference between any two consecutive numbers in an AP is the common, and the difference is known as the common difference.

• Even number: The numbers that are divisible by 2 are even numbers, for example, 2, 4, 6, 8.