Sum of First n 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.

Let us check the above formula with some examples.

n = 1, S1 = 1 * (1 + 1) / 2 = 1. The first natural number is 1 only, so sum would be 1.
n = 2, S2 = 2 * (2 + 1)/ 2 = 3. The first two natural numbers are 1 and 2. If we sum them we get 3 only
n = 3, S2 = 3 * (3 + 1)/ 2 = 6 The first two natural numbers are 1, 2 and 3. If we sum them we get 6 only

Poof 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 Series
• n is Number of Terms in Series = 100
• A₁ is First Term= 1
• A_n is 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)

Articles Related to Sum of Natural Numbers,

• Program for Sum of Natural Numbers
• Triangular Number Sequence
• Sum in Maths
• Arithmetic Operations
• Arithmetic Progression and Geometric Progression

Examples on Sum of Natural Numbers

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

Solution:

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:

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

Q1. Find the sum of the first 5 even natural numbers.

Q2. Find the sum of the first 5 odd natural numbers.

Q3. Find the sum of the first 20 natural numbers.

Q4. Name some of the common points based on the Sum of natural numbers.