Sum of the series 1 + (1+2) + (1+2+3) + (1+2+3+4) + …… + (1+2+3+4+…+n)

Given the value of n, we need to find the sum of the series where i-th term is sum of first i natural numbers.

Examples :

Input  : n = 5   
Output : 35
Explanation :
(1) + (1+2) + (1+2+3) + (1+2+3+4) + (1+2+3+4+5) = 35

Input  : n = 10
Output : 220
Explanation :
(1) + (1+2) + (1+2+3) +  .... +(1+2+3+4+.....+10) = 220

Naive Approach :
Below is implementation of above series :

C++

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// CPP program to find sum of given series
#include <bits/stdc++.h>
using namespace std;
  
// Function to find sum of given series
int sumOfSeries(int n)
{
    int sum = 0;
    for (int i = 1 ; i <= n ; i++)
        for (int j = 1 ; j <= i ; j++)
            sum += j;
    return sum;
}
  
// Driver Function
int main()
{
    int n = 10;
    cout << sumOfSeries(n); 
    return 0;
}

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Java

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// JAVA Code For Sum of the series
import java.util.*;
  
class GFG {
      
    // Function to find sum of given series
    static int sumOfSeries(int n)
    {
        int sum = 0;
        for (int i = 1 ; i <= n ; i++)
            for (int j = 1 ; j <= i ; j++)
                sum += j;
        return sum;
    }
      
    /* Driver program to test above function */
    public static void main(String[] args) 
    {
         int n = 10;
         System.out.println(sumOfSeries(n)); 
          
    }
}
  
// This code is contributed by Arnav Kr. Mandal.

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Python

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# Python3 program to find sum of given series 
  
# Function to find sum of series
def sumOfSeries(n):
    return sum([i*(i+1)/2 for i in range(1, n + 1)])
  
# Driver Code 
if __name__ == "__main__":
    n = 10
    print(sumOfSeries(n))

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C#

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// C# Code For Sum of the series
using System;
  
class GFG {
  
    // Function to find sum of given series
    static int sumOfSeries(int n)
    {
        int sum = 0;
        for (int i = 1; i <= n; i++)
            for (int j = 1; j <= i; j++)
                sum += j;
        return sum;
    }
  
    /* Driver program to test above function */
    public static void Main()
    {
        int n = 10;
          
        Console.Write(sumOfSeries(n));
    }
}
  
// This code is contributed by vt_m.

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PHP

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<?php
// PHP program to find 
// sum of given series
  
// Function to find 
// sum of given series
function sumOfSeries($n)
{
    $sum = 0;
    for ($i = 1 ; $i <= $n ; $i++)
        for ($j = 1 ; $j <= $i ; $j++)
            $sum += $j;
    return $sum;
}
  
// Driver Code
$n = 10;
echo(sumOfSeries($n)); 
  
// This code is contributed by Ajit.
?>

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Output :

220



Efficient Approach :

Let n^{th} term of the series 1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4)…(1 + 2 + 3 +..n) be denoted as an

an = Σn1 i = \frac{n (n + 1)}{2} = \frac{(n^2 + n)}{2}

Sum of n-terms of series 
Σn1 an = Σn1 \frac{(n^2 + n)}{2} 

      = \frac{1}{2} Σ  [   n^2  ]  + Σ  [   n    ] 

      = \frac{1}{2} * \frac{n(n + 1)(2n + 1)}{6} + \frac{1}{2} * \frac{n(n+1)}{2}

      = \frac{n(n+1)(2n+4)}{12}

Below is implementation of above approach :

C++

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// CPP program to find sum of given series
#include <bits/stdc++.h>
using namespace std;
  
// Function to find sum of given series
int sumOfSeries(int n)
{
    return (n * (n + 1) * (2 * n + 4)) / 12;
}
  
// Driver Function
int main()
{
    int n = 10;
    cout << sumOfSeries(n); 
}

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Java

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// JAVA Code For Sum of the series
import java.util.*;
  
class GFG {
      
    // Function to find sum of given series
    static int sumOfSeries(int n)
    {
        return (n * (n + 1) * 
                (2 * n + 4)) / 12;
    }
      
    /* Driver program to test above function */
    public static void main(String[] args) 
    {
         int n = 10;
         System.out.println(sumOfSeries(n)); 
          
    }
}
  
// This code is contributed by Arnav Kr. Mandal.

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Python

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# Python program to find sum of given series
  
# Function to find sum of given series
def sumOfSeries(n):
    return (n * (n + 1) * (2 * n + 4)) / 12;
      
# Driver function
if __name__ == '__main__':
    n = 10
    print(sumOfSeries(n))

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C#

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// C# Code For Sum of the series
using System;
  
class GFG {
  
    // Function to find sum of given series
    static int sumOfSeries(int n)
    {
        return (n * (n + 1) * (2 * n + 4)) / 12;
    }
  
    /* Driver program to test above function */
    public static void Main()
    {
        int n = 10;
          
        Console.Write(sumOfSeries(n));
    }
}
  
// This code is contributed by vt_m.

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PHP

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<?php
// PHP program to find
// sum of given series
  
// Function to find 
// sum of given series
function sumOfSeries($n)
{
    return ($n * ($n + 1) * 
           (2 * $n + 4)) / 12;
}
  
// Driver Code
$n = 10;
echo(sumOfSeries($n)); 
  
// This code is contributed by Ajit.
?>

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Output :

220


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