Examples Exercises On Natural Numbers 

What are referred to as natural numbers?

In mathematics, natural number systems can be referred to as a part of a number system constituting all positive integers starting from 1 to infinity. These numbers do not include zero and are used for counting purposes. Therefore, the natural numbers, 1, 2, 3, 4, 5, 6, 7, 8,……., can also be called counting numbers. 

Natural numbers and Whole Numbers 

In mathematics, it is said that all natural numbers are whole numbers, but all whole numbers are not natural numbers. This can be proven by the fact that natural numbers include all whole numbers but not zero. So, it can be said that all natural numbers are whole numbers.

On the other hand, whole numbers include zero along with other natural numbers. Therefore, it can be said that not all whole numbers are natural numbers.

Natural numbers: {1, 2, 3, 4, 5, 6,……..} 

Whole Numbers: {0, 1, 2, 3, 4, 5, 6,}

What is meant by the sum of the natural number formula?

5 + 10 + 15 + 20 + 25 +….. up to x terms. This particular sequence is arranged in the arithmetic series. Hence, the arithmetic progression sum of x terms formula may be used to obtain the formula for the sum of the first x integers. 

Sum of first x numbers = ∑1x = [x (x + 1)] / 2

Here, x represents the natural number.

When talking about the definition of the sum of the first x natural numbers formula, it can be defined as the form of arithmetic progression where the sum of x terms can be arranged in a sequence with 1 being the first term, x being the total number of terms along with the nth term.

Examples:

Q.1 Find the sum of the first 36 natural numbers.

Solution: The mathematical formula for calculating the sum of first x natural numbers is:

∑x = [x (x + 1)] / 2

Here, x = 36

Put x = 36 in the formula:

∑36 = [36 (36 + 1)] / 2

∑36 = [36 (37)] / 2

∑36 = [1332] / 2

∑36 = 666

What is meant by the sum of squares of x natural numbers?

Here the sum of squares of x natural numbers implies the addition of squared numbers starting from 1 to infinity. The squared terms can be first x even natural numbers, first x odd natural numbers, first x consecutive natural numbers, etc.; now that the sum of first x natural number is represented as ∑x, the sum of squares of x natural number can be represented as ∑x2.

The sum of squares of x natural numbers can be calculated using the following mentioned formula:

Sum of squares of x natural numbers = ∑x2 = [x (x + 1)(2x + 1)] / 6

Apart from the above-mentioned formula, there are two other formulas used to calculate the sum of squares of first x even natural numbers and the sum of squares of first x odd natural numbers, respectively.

Sum squares of first x even natural numbers = ∑x2 = [2x (x + 1)(2x + 1)] / 3

Sum squares of first x odd natural numbers = ∑x2 = [x (2x + 1)(2 – 1)] / 3 

Examples:

Q.1 Find the sum of squares of the first 55 natural numbers.

Solution: The mathematical formula for calculating the sum of squares of first x natural numbers is:

∑x2 = [x (2x + 1)(2 – 1)] / 6

Put x = 55 in the formula:

Here, x = 55

∑552= [55 (55 + 1)(2 x 55 + 1)] / 6

∑552= [55 (56)(111)] / 6

∑552= [55 (6216)] / 6

∑552= [341880] / 6

∑552= 56980

Q.2 Find the sum of squares of the first 20 even natural numbers.

Solution: The mathematical formula for calculating the sum of squares of first x even natural numbers is:

 ∑x2 = [2x (x + 1)(2x + 1)]/ 3

Here, x = 20

Put x = 20 in the formula:

∑202 = [2 x 20 (20 + 1)(2 x 20 + 1)] / 3

∑202 = [40 (21)(41)] / 3

∑202 = [40 (861)] / 3

∑202 = [34440] / 3

∑202 = 11480

Q.3 Find the sum of squares of the first 15 odd natural numbers.

Solution: The mathematical formula for calculating the sum of squares first x odd natural numbers is:

∑x2 = [x (2x + 1)(2x – 1)] / 3

Here, x = 15

Put x = 15 in the formula:

∑152 = [15 (2 x 15 + 1)(2 x 15 – 1)] / 3

∑152 = [15 (31)(29)] / 3

∑152 = [15 (899)] / 3

∑152 = [13485] / 3

∑152 = 4495

Conclusion

Natural numbers form an important part of the number system in mathematics. This particular set of numbers in the number system consists of positive integers that start from 1 to infinity. Now that zero is not included in the natural numbers, but in whole numbers, it can be said that all natural numbers are whole numbers. To know more about the concept of natural numbers and their formulas, it is recommended to go through the above-mentioned article.