Sum of first n natural numbers formula and derivation

Learn how to find sum of first n natural numbers. Here am going to give information about the method for finding n natural numbers sum. 

Learn how to find sum of first n natural numbers. Here am going to give information about the method for finding n natural numbers sum.

Derivation of Formula for finding sum of n natural numbers:

Let

 ​​ ​​ ​​ ​​ ​​​​ sn​​ = sum of n terms

 ​​ ​​ ​​ ​​ ​​ ​​​​ n = number of terms

sn=   1 +  ​​ ​​​​ 2 +  ​​ ​​ ​​ ​​ ​​​​ 3 +  ​​ ​​ ​​ ​​ ​​ ​​​​ 4 +  ​​ ​​ ​​ ​​ ​​ ​​​​ 5 +……………………+ (n-4) + (n-3) + (n-2) + (n-1) + n   ​​ eq 1

sn= ​​ ​​​​ n + (n-1) + (n-2) + (n-3) + (n-4)……………  ​​ ​​​​ ……  ​​ ​​​​ +5 +  ​​ ​​ ​​​​ 4  ​​ ​​​​ +  ​​ ​​​​ 3  ​​ ​​​​ +  ​​ ​​ ​​​​ 2  ​​​​ + 1  ​​ ​​​​ ​​ eq 2

Add eq 1 and 2

We get

2​​ sn​​ = (n+1) + (n+1) + (n+1) + (n+1) + (n+1) +………………..+ (n+1) + (n+1) + (n+1) + (n+1) + (n+1)​​ 

2​​ sn​​ = n​​ ×​​ ​​ ( n+1)  ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ (​​ ​​ we are adding ( n+1)  ​​​​ n times)

sn​​ =​​ n (n+1)2 ​​​​ 

 

Examples on sum of first n natural numbers:

  1. Find the sum of first 100 natural numbers.

Solution:

 ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ sn​​ =​​ n (n+1)2 ​​​​ 

 ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ =​​ 100 ( 100+1)2​​ 

 ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ = 50​​ ×​​ 101

 ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ = 5050

 

2. Find the sum of first 50 natural numbers?

Solution:

                    sn​​ =​​ n (n+1)2 ​​​​ 

 ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ =​​ 50 ( 50+1)2​​ 

 ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ = 25​​ ×​​ 51

 ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ = 1275

 

If you have any doubts about sum of first n natural numbers, then let me know.

 

 

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