# 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:

- Decimal Number System
- Binary Number System
- Octal Number System
- 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 × 10

^{2}) + (5 × 10^{1}) + (3 × 10^{0})= (1 × 100) + (5 × 10) + (3 × 1) { where, 10

^{0}= 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 × 2^{5}+ 0 × 2^{4}+ 0 × 2^{3}+ 1 × 2^{2 }+ 1 × 2^{1}+ 1 × 2^{0}= 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 × 8

^{2 }+ 4 × 8^{1 }+ 5 × 8^{0}= 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 × 16

^{3}+ 6 × 16^{2}+ 11 × 16^{1}+ 12 × 16^{0}= 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 = 64Therefore, the sum of first 8 odd natural numbers is 64.

**Alternate Method**

We know that the sum of n odd numbers is equal to N^{2} .

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

According to the question here N = 8

=> N

^{2 }=> (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 (N^{2 }= 10 × 10 = 100)

2. Sum of first 3 odd natural numbers = 1 + 3 + 5 = 9 (N^{2 }= 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:

N

^{2}= 50 × 50 => 2500or

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

=> N

^{2 }= 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.

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