# Sum of sum-series of first N Natural numbers

Last Updated : 25 Jul, 2022

Given a natural number n, find the sum of the sum-series of the first N natural number.

Sum-Series: is sum of first N natural numbers, i.e, sum-series of 5 is 15 ( 1 + 2 + 3 + 4 + 5 ).

Example:

Input: N = 5
Output: 35
Explanation:
Sum of sum-series of {1, 2, 3, 4, 5} i.e. {1 + 3 + 6 + 10 + 15} is 35.

Input: N = 2
Output:
Explanation:
Sum of sum-series of {1, 2} i.e. {1 + 3} is 4.

Simple approach:
Find sum series for every value from 1 to N and then add it.

• Create a variable Total_sum to store the required sum series.
• Iterate over the number from 1 to N
• Find sum-series of every value by using the formulae sum = (N*(N + 1)) / 2
• Add the value to Total_sum
• In the end, print the value stored in Total_sum

Below is the implementation of the above approach:

## C++

 // C++ program to implement// the above approach#includeusing namespace std; // Function to find the sumstatic long sumOfSumSeries(int N){    long sum = 0L;     // Calculate sum-series    // for every natural number    // and add them    for (int i = 1; i <= N; i++)     {        sum = sum + (i * (i + 1)) / 2;    }     return sum;} // Driver codeint main(){    int N = 5;    cout << sumOfSumSeries(N);} // This code is contributed by Code_Mech

## Java

 // Java program to implement// the above approach class GFG {     // Function to find the sum    static long sumOfSumSeries(int N)    {         long sum = 0L;         // Calculate sum-series        // for every natural number        // and add them        for (int i = 1; i <= N; i++) {            sum = sum + (i * (i + 1)) / 2;        }         return sum;    }     // Driver code    public static void main(String[] args)    {        int N = 5;        System.out.println(sumOfSumSeries(N));    }}

## Python3

 # Python3 program to implement # the above approach  # Function to find the sum def sumOfSumSeries(N):      _sum = 0     # Calculate sum-series     # for every natural number     # and add them     for i in range(N + 1):         _sum = _sum + (i * (i + 1)) // 2     return _sum  # Driver code N = 5 print(sumOfSumSeries(N))      # This code is contributed by divyamohan123

## C#

 // C# program to implement// the above approachusing System;class GFG{ // Function to find the sumstatic long sumOfSumSeries(int N){    long sum = 0L;     // Calculate sum-series    // for every natural number    // and add them    for(int i = 1; i <= N; i++)     {       sum = sum + (i * (i + 1)) / 2;    }         return sum;} // Driver codepublic static void Main(){    int N = 5;         Console.Write(sumOfSumSeries(N));}} // This code is contributed by Nidhi_Biet

## Javascript

 

Output
35

Time complexity: O(N)
Auxiliary Space: O(1)

Efficient approach:
Total_sum of the above series can be calculated directly by using the below formulae:

where N is the natural number

Proof of the above formula:
Lets assume N = 5

1. Then the sum is sum of all the below elements in the table, let’s call this “result”

let’s populate the empty cells with the same value in other columns, lets’s call this “totalSum

1.

As sum of N numbers is repeated N times
totalSum = N * [(N*(N + 1))/2]
populated data = (1 times * 2) + (2 times * 3) + (3 times * 4) + (4 times * 5)
= 1*2 + 2*3 + 3*4 ……… +(N-1)*N
=[(N-1) * (N) * (N+1)]/3

1. Since,

result = totalSum – populatedData
= N * [(N*(N+1))/2] – [(N-1) * (N) * (N+1)]/3
= (N*(N+1)*(N+2))/6

1. Therefore

Below is the implementation of the above approach:

## C++

 // C++ program to implement// the above approach#include #include using namespace std; // Function to find the sumlong sumOfSumSeries(int n){    return (n * (n + 1) * (n + 2)) / 6;} // Driver codeint main (){    int N = 5;    cout << sumOfSumSeries(N);    return 0;} // This code is contributed// by shivanisinghss2110

## Java

 // Java program to implement// the above approach class GFG {     // Function to find the sum    static long sumOfSumSeries(int n)    {        return (n * (n + 1) * (n + 2)) / 6;    }     // Driver code    public static void main(String[] args)    {        int N = 5;        System.out.println(sumOfSumSeries(N));    }}

## Python3

 # Python3 program to implement # the above approach # Function to find the sum def sumOfSumSeries(n):          return (n * (n + 1) * (n + 2)) // 6 # Driver code N = 5 print(sumOfSumSeries(N)) # This code is contributed by divyamohan123

## C#

 // C# program to implement the // above approachusing System;class GFG{ // Function to find the sumstatic long sumOfSumSeries(int n){    return (n * (n + 1) * (n + 2)) / 6;} // Driver codepublic static void Main(String[] args){    int N = 5;         Console.Write(sumOfSumSeries(N));}} // This code is contributed by Ritik Bansal

## Javascript

 

Output
35

Time complexity: O(1), considering multiplication, addition & division takes constant time.
Auxiliary Space: O(1)