Find the sum of first 8 odd numbers

The method to represent and work with numbers is known as number system. A number system is a system of writing to represent numbers. It is the mathematical notation used to represent numbers of a given set by using digits or other symbols. It allows us to operate arithmetic operations such as division, multiplication, addition, subtraction.

Some important number systems are as follows:

1. Decimal Number System
2. Binary Number System
3. Octal Number System
4. Hexadecimal Number System

Let’s see about all these number systems in detail.

Decimal Number System

The decimal number system consists of ten digits i.e. from 0 to 9. The base of decimal number system is 10. These digits can be used to represent or express any numeric value.  

For example, the decimal number 153 consists of the digit 3 in ones place, the digit 5 in the tens place, and the digit 1 in hundreds place which can be represented as:

(1 × 10²) + (5 × 10¹) + (3 × 10⁰)

= (1 × 100) + (5 × 10) + (3 × 1)                                       { where, 10⁰ = 1}

= 100 + 50 + 3

= 153

Binary Number System

The binary number system consists of only two digits i.e. 0 and 1. The base of the binary number system is 2. The digital computer represents all kinds of data in a binary number system.

For example, convert 100111 into a decimal number system.

(100111)₂  = 1 × 2⁵ + 0 × 2⁴ + 0 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰

                 = 32 + 0 + 0 + 4 + 2 + 1

                 = (39)₁₀

Octal Number System

The octal number system consists of digits from 0 to 7. The base of octal number system is 8. Octal number systems are basically used in computer applications.

For example, convert 1458 into decimal.

1458  = 1 × 8² + 4 × 8¹ + 5 × 8⁰

          = 64 + 32 + 5

          = 10110

Hexadecimal Number System

In the hexadecimal number system, numbers are first represented from digits 0 to 9 as decimal number system and then the numbers are represented using alphabets from A to F. The base of the hexadecimal number system is 16.

For example, convert 26BC16 to decimal.

26BC16  = 2 × 16³  + 6 × 16² + 11 × 16¹ + 12 × 16⁰

              = 8192 + 1536 + 176 + 12

              = 991610

What are odd numbers?

Odd numbers are that numbers that are not completely divisible by 2 and hence give the remainder as 1 are called an odd number. For example 1, 3, 5, 7, 9, 11 and so on. In other words, odd numbers are the numbers that give the remainder as 1.

Explanation:

• 1/2 = 1        (Odd Number)
• 2/2 = 0        (Even Number)
• 3/2 = 1        (Odd Number)
• 4/2 = 0        (Even Number)
• 5/2 = 1        (Odd Number)
• 6/2 = 0        (Even Number)
• 7/2 = 1        (Odd Number)
• 8/2 = 0        (Even Number)
• 9/2 = 1        (Odd Number)
• 10/2 = 0      (Even Number)

According to the question, if we need to find the first 8 odd numbers, we need to divide every number starting from 1 by 2 and find all the odd numbers. 

Therefore first 8 odd natural numbers are 1, 3, 5, 7, 9, 11, 13, 15. As we see that it is forming an arithmetic progression where the common difference d is 2 and the first term is 1 and the last term is 15.

Now, we know that sum is given by S = n/2 [2a + (n−1) d]

here:

• n = 8  (number of digits in the series)
• a = 1  (First term of an A.P)
• d= 2  (Common difference in an A.P)

By substituting the values we get,

S = 8/2 [2 × 1 + (8 – 1) × 2]

S = 4 [2 + 7 × 2]

S = 4 × 16

S = 64

Therefore, the sum of first 8 odd natural numbers is 64.

Alternate Method

We know that the sum of n odd numbers is equal to N² . 

Sum of first 8 odd numbers = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64

According to the question here N = 8 

=> N² 

=> (8)²

=> 8 × 8

=> 64

This is very effective approach to find the sum of n odd numbers.

Some more examples are 

1. Sum of first 10 odd natural numbers = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100  (N² = 10 × 10 = 100)

2. Sum of first 3 odd natural numbers = 1 + 3 + 5 = 9  (N² = 3 × 3 = 9)

Similar Questions

Question 1: What is the sum of odd natural numbers from 1 to 100?

Answer: 

We know that there are 50 odd natural numbers between 1 to 100. Therefore, here n = 50

The sum of odd natural numbers between 1 to 100 is 2500.

Explanation:

N² = 50 × 50  => 2500

or 

S = n/2 [ 2a + (n−1) d ]

S = 50/2 [ 2 × 1 + (50 – 1) × 2 ]

S = 25 [ 2 + 49 × 2 ]

S = 25 × 100

S = 2500 

Question 2. What is the sum of the first 12 odd natural numbers?

Answer: 

Sum of first 12 odd natural numbers are = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 = 144 

Explanation:

here n= 12 

=> N² = 12 × 12 

=> 144 

or

S = n/2 [2a + (n−1) d]

S = 12/2 [2 × 1 + (12 – 1) × 2]

S = 6 [2 + 11 × 2]

S = 6 × 24

S = 144 

Hence, the sum of the first 12 odd natural number is 144.