In this article, you will read the Definition, Formula, sum and solution of important questions of Natural Numbers.

On the previous page, we have shared the post of **Number System**, an important chapter of mathematics, so definitely read the square root.

Let us read and understand the definition, formulas, tricks, and examples of natural numbers on this page.

## What are Natural Number

All Numbers used in Counting are called Natural Numbers.

**Ex :** 1, 2, 3, 4, 5, 6, 7, 8, 9………

Natural Numbers are Denoted by N.

The Numbers which are used to count an object are called natural numbers.

**As :-**

- There are 5 books kept in the bag.
- There are 4 pens on the table.
- There are 30 students studying in a Mathematics class.
- Natural numbers are also used for ordering.

**As :-**

- Ram stood first in class 10th.
- Sita secured 4th rank in Law.
- Shyamu sits sixth in the class.

## Formulas for Natural Numbers

- Average of first n natural odd numbers = n
- Average of first n natural even numbers = n + 1
- Sum of natural numbers = (first number + last number / 2) × n
- Average of first n natural numbers = ( n + 1 ) / 2
- Average of n consecutive odd natural numbers = (n + 1) / 2
- Sum of consecutive odd natural numbers up to n = (n/2 + 1)
- N = (last number – first number / class interval) + 1
- Sum of squares of first n natural numbers = n( n + 1 )( 2n + 1 ) / 6
- Sum of cubes of first n natural numbers = [ n (n + 1) / 2 ]²

## Important Point :

- The smallest natural number is 1.
- The largest natural number is infinity.
- In the set, 0 is considered a natural number.
- Natural number are a part of number system.
- Natural number are represented by N.
- Where are natural number known as Hindi Arabic numbers on the basis of ancient history.
- Natural numbers keep on increasing by 1 unit continuously.
- Natural numbers = 0 < N ≤ 1.
- The set is denoted by N0 = {1, 2, 3, 4, 5, 6, 7, 8, 9, ……).

## Natural Number From 1 to 100

## Sum of First 10 Natural Number

First 10 natural number = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Sum of first 10 natural number = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

Sum of first 10 natural number = 55

Hence the sum of first 10 natural number is 55.

**Definition of Square : **When a number is multiplied by the same number, the product obtained is called the square of the number.

**What is the average of the squares of the first 10 natural number?**

## Square from 1 to 10

1² | 1 |

2² | 4 |

3² | 9 |

4² | 16 |

5² | 25 |

6² | 36 |

7² | 49 |

8² | 64 |

9² | 81 |

10² | 100 |

Sum of squares of first 10 natural number = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100

Sum of squares of first 10 natural number = 385

## Questions Based on Natural Number

Q.1 1 + 2 + 3 + 4 ……………… 18 + 19 + 20 = ?

A. 120

B. 210

C. 240

D. 320

Solution : According to the question,

1 + 2 + 3 + 4 ……………… 18 + 19 + 20 = ?

last number = 20

first number = 1

N = (last number – first number)/class interval + 1

N = ( 20 – 1 ) / 1 + 1

N = 19 + 1

N = 20

Sum = (first number + last number / 2) × n

Sum = (1 + 20 ) / 2 ×20

Sum = 21 × 20 / 2

sum = 21 × 10

sum = 210

Ans. 210

Q.2 1+2+3+4 ……………… 78+79+80 = ?

A. 1,220

B. 1,880

C. 3,240

D. 4,860

Solution : According to the question,

1+2+3+4 ……………… 78+79+80 = ?

last number = 80

first number = 1

N = (last number – first number)/class interval + 1

N = ( 80 – 1 ) / 1 + 1

N = 79 + 1

N = 80

Sum = (first number + last number / 2) × n

Sum = (1 + 80 ) / 2 × 80

Sum = 81 × 80 / 2

sum = 81 × 40

sum = 3,240

Ans. 3,240

Q.3 What is the sum of all the odd numbers from 1 to 49?

A. 360

B. 625

C. 720

D. 800

Solution : According to the question,

1 + 3 + …………….………+ 48 + 49

first number = 1

last number = 49

N = (last number – first number)/class interval + 1

N = (49 – 1)/2 + 1

N = 48/2 + 1

N = 24 + 1

N = 25

Sum of natural number = (first number + last number)/2 × n

Sum = (1 + 49)/2 × 25

Sum = 50/2 × 25

Sum = 25 × 25

sum = 625

Ans. 625

Q.4 What is the sum of squares of all natural number from 1 to 20?

A. 1,385

B. 4,350

C. 2,260

D. 2,870

Solution : According to the question,

n = 20

Sum = n (n + 1) (2n + 1) / 6

Sum = 20 (20 + 1) (2 × 20 + 1) / 6

Sum = 20 × 21 (40 + 1) / 6

Sum = (20 × 21 × 41) / 6

Sum = 17,220 / 6

sum = 2,870

Ans. 2,870

Q.5 What is the sum of squares of all natural number from 1 to 80?

A. 1,37,895

B. 1,24,350

C. 2,26,670

D. 1,73,880

Solution : According to the question,

n = 80

Sum = n (n + 1) (2n + 1) / 6

Sum = 80 (80 + 1) (2 × 80 + 1) / 6

Sum = 80 × 81 (160 + 1) / 6

Sum = (80 × 81 × 161) / 6

Sum = 1,043,280 / 6

Sum = 1,73,880

Ans. 1,73,880

Q.6 What is the sum of cubes of all natural numbers from 1 to 40?

A. 468,900

B. 672,400

C. 678,908

D. 566,902

Solution : According to the question,

[n (n + 1) / 2]²

n = 40

[40 (40 + 1) / 2]²

(40 × 41 / 2)²

(1,640 / 2)²

(820)²

Ans. 672,400

Q.7 The number obtained by subtracting four times the number from thrice the square of a natural number is 50 more than that number. is that number?

A. 4

B. 5

C. 6

D. 10

Solution : According to the question,

Let x be a natural number.

Then, x² × 3 – 4x = x + 50

3x² – 4x – x = 50

3x² – 5x = 50

3x² – 5x – 50 = 0

3x² + 10x – 15x – 50 = 0

x (3x + 10) – 5 (3x + 10) = 0

(x – 5) (3x + 10) = 0

x – 5 = 0 , 3x + 10 = 0

x = 5 or 3x = – 10

x = 5 or x = – 10/3 (not natural number.

Ans. x = 5 is a natural number.

Q.8 On subtracting the square of a natural number from its cube, the result is 48. will be the number?

Solution : According to the question,

Let x be a natural number.

x³ – x² = 48

x² (x – 1) = 48

x² (x – 1) = 16 × 3

4² × (4 – 1)

x = 4

Q.9 Subtracting the square of a natural number from its cube gives 100. is that number?

Solution : According to the question,

Let x be a natural number.

x³ – x² = 100

x² (x – 1) = 100

x² (x – 1) = 25 × 4

5² × (5 – 1)

x = 5

Q.10 What is the sum of cubes of all natural number from 1 to 100?

A. 12,867,900

B. 18,234,400

C. 27,856,908

D. 25,502,500

Solution : According to the question,

[n (n + 1) / 2]²

n = 100

[100 (100 + 1) / 2]²

[(100 × 101) / 2]²

(10,100 / 2)²

(5,050)²

Ans. 25,502,500

### FAQ

**1. What are the first twenty natural number?**

**2. Is every whole number a natural number?**

**Answer : **All whole number except zero are natural number. Because the value of a whole number ranges from zero to infinity. The same natural number range from one to infinity.

**3. How many natural number are there between 25 and 36?**

**Answer : **There are 10 natural number between 25 and 36. (26, 27, 28, 29, 30, 31, 32, 33, 34, 35)

**4. Find the average of natural number from 1 to 100?**

**Answer : **Average of numbers = Sum of total numbers / Total numbers

Average of numbers = (1+100)/2

= 101/2

= 50.5

Ans. 50.5

**5. Find the sum of natural number from 1 to 100?**

**Answer : **Sum of natural number = Average of number × number

Sum of natural number = 50.5×100

= 5050

Ans. 5050

**6. Average of even numbers from 1 to 100?**

**Answer : **Average of the numbers = Sum of the total numbers × Total numbers

Average of numbers = (2+100)/2

= 102/2

= 51

Ans. 51

**7. Sum of even natural numbers from 1 to 100?**

**Answer : **Sum of natural number = Average × Total number

Sum of natural even number = 51 × 51

= 2550

Ans. 2550

**8. Sum and average of odd numbers from 1 to 100?**

**Answer : **Average of odd numbers = Sum of total numbers / Total numbers

Average of odd numbers = sum of numbers × total numbers

Average of odd numbers = (1 + 99)/2

= 100/2

= 50

Sum of odd numbers = 50×50

= 2500

Ans. 2500

**9. What is the smallest natural number?**

**Answer : **The smallest natural number is 1.

**10. Which is the largest natural number?**

**Answer : **There is no largest natural number. Because the natural number range from 1 to infinity.

**Must Read :**

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