Sum of Natural Numbers

Sum of n Natural Numbers is simply an addition of ‘n’ numbers of terms that are organized in a series, with the first term being 1, and n being the number of terms together with the nth term. The numbers that begin at 1 and terminate at infinity are known as natural numbers. Sum of the first n natural numbers formula is given by [n(n+1)]/2.

In this article, we will learn about the sum of natural numbers, sum of natural number definition, sum of natural number formula, derivation of sum of natural number formula and solve some examples.

What is Sum of Natural Numbers?

Sum of Natural Numbers is the addition of Natural Numbers starting from 1. Natural Numbers are the numbers that begin with 1, a positive integer, and go on with infinity numbers, whereas they don’t include zero or any negative numbers in their sequence of numbers. These numbers help us to do various arithmetic calculations in day-to-day activities.

Sum of the First n Natural Numbers

In the Sum of natural numbers, the arithmetic calculations are based on adding consecutive numbers together or adding any of the natural numbers together. Even though, there is a consequence that if we add any two natural numbers together, we will get the natural number only. Let’s see a few examples of these terms.

• 3 + 2 = 5
• 5 + 6 = 11

Sum of Natural Numbers Formula

Sum of first n natural numbers can be found using a simple formula:

S_n = n × (n + 1) /2

• S_n is the sum of the first n natural numbers.
• n is the number of natural numbers.

Derivation of Sum of Natural Numbers Formula

All natural numbers continue till infinity and so on. If students examine closely, they can find an arithmetic progression by using a formula in the following ways:

Let’s derive the formula for finding the number of terms (n) in an arithmetic series using the given formula

where, n is the number of terms, A₁ is the first term, A_n is the last term.

• So, Let’s see the form of an arithmetic series is:
  A_n = A₁ + (n – 1) × Common difference
• Now, to find the number of terms (n). Let’s rearrange the formula for A_n to solve for n]:
  n = (A_n – A₁ / common difference) + 1

Let’s see the simple formula derivation of the Sum of natural numbers

• In, the very first step Consider the Sum
  S_n = 1 + 2 + 3 + … + (n−2) + (n−1) + n
• Now, Let’s pair the terms each pair sums to n+1:
  S_n = (1 + n) + (2 + (n−1)) + (3 + (n−2)) +…
• Number of Pairs: There are n terms in the sum, and each pair contributes n+1. Therefore, the number of pairs is n/2.
• Write the Sum as Products: Express the sum as the product of the number of pairs and the constant sum of each pair:
• S_n = n/2 × (n+1)
• At last, let’s simplify it, S_n = n/2 × (n+1). If you multiply n/2 by (n+ 1) you get n × (n + 1) /2
  Then, we get the sum of natural number formula: S_n = n × (n + 1) /2

Sum of Natural Numbers 1 to 100

To find the sum of natural numbers from 1 to 100, you can use the formula for the sum of an arithmetic series. The formula is:

S_n = n/2 × (A₁+ A_n)

Here, S_n is a Sum of the series.

n is the number of terms in the series = 100

A₁ is the first term= 1

A_n is the last term= 100

In this case, you want the sum of natural numbers from 1 to 100, so n= 100, A₁ = 1, and A_n = 100. Let’s use the formula, and put these values in it:

S_n = 100/2 × (1+100)

S_n= 50 × 101

S_n= 5050

Therefore, the sum of natural numbers from 1 to 100 is 5050.

Sum of Natural Numbers Starting from 1

The sum of natural numbers starting from 1 formula is given by:

S_n = n/2 × (A₁+ A_n)

putting A₁ = 1

S_n = n/2 × (1 + A_n)

Sum of Natural Numbers Not Starting from 1

The sum of natural numbers not starting from 1 formula is given by:

S_n = n/2 × (A₁+ A_n)

Read More About,

• Natural Numbers
• Arithmetic Progression and Geometric Progression

Solved Examples on Sum of Natural Numbers

Example 1: Determine the sum of natural numbers between 100 and 150.

Solution:

The Sum in arithmetic Series: S_n is the sum of the series; n is the number of terms in the series that can be used by using formula:

n = (A_n – A₁/ common difference) + 1

A₁ is the first term = 101

A_n is the last term = 149

Common difference = 1

Let’s calculate the value of n by using formula:

n = (A_n – A₁/ common difference) + 1

n = (149- 101 / 1) + 1

n = 49 + 1

n = 50

Now, we will calculate the sum of natural numbers:

S_n = n/2 × (A₁ + A_n)

S_n = 50/2 × (101 + 149)

S_n = 25 × 250

S_n = 6250

Therefore, the sum of natural numbers between 100 and 150 using the formula is 6250.

Example 2: Add any two Consecutive natural numbers, which is always an odd number. Justify this statement.

Solution:

When you add any two consecutive natural numbers, n and n+1, the sum is 2n+1. The term 2n represents an even number since it’s divisible by 2, and adding 1 to an even number always results in an odd number. Therefore, the sum of consecutive natural numbers is consistently an odd number.

For Example:

Let’s add any two consecutive natural numbers: 3 and 4

3+4 = 7

Now, using the formula 2n + 1, where n is the first natural number: 2 × 3 + 1 = 7

Hence, adding the consecutive natural numbers 3 and 4 results in the sum 7, which is an odd number.

Example 3: Find the sum of the first 5 natural odd numbers using the formula Sn = n/2 × (A₁ + A_n).

Solution:

The Sum in arithmetic Series:

S_n is a Sum of the series; n is the number of terms in the series = 5

A₁ is the first term = 1

A_n is the last term = 9

In this case, you want the sum of the first 5 natural odd numbers so n= 5, A₁ = 1, and A_n = 9. Let’s use the formula, and put these values in it:

S_n = 5/2 × (1+9)

S_n = 5/2 × 10

S_n = 25

Therefore, the sum of the first 5 natural odd numbers is 25.

Example 4: Find the sum of the first 30 natural numbers.

Solution:

Let’s see the sum of first 30 natural numbers

S = [n(n+1)]/2

S = [30 (30+1)]/2

S = 465

Sum of Natural Numbers – Practice Questions

1. Find the sum of the first 5 even natural numbers.

2. Find the sum of the first 5 odd natural numbers.

3. Find the sum of the first 20 natural numbers.

4. Name some of the common points based on the Sum of natural numbers.

Sum of Natural Numbers – FAQs

What is the Formula for the Sum of Natural Numbers?

The sum of natural numbers can be found using the formula S_n = n × (n + 1) /2.

Why is the Sum of Natural Numbers Important?

The sum of natural numbers can be used to solve a variety of mathematical and practical issues, including series, discrete quantity computations, and arithmetic progressions.

What is the Sum of First 100 Natural Numbers?

Sum of first 100 Natural Numbers is 5050

What is the Sum of First 10 Natural Numbers?

The sum of first 10 natural numbers is given by:

S = n × (n + 1) /2 = 10 × 11 /2 = 55

How to Find the Sum of n Natural Numbers?

The sum of natural numbers can be obtained by using formula: S_n = n × (n + 1) /2