Sum of the series 1 + (1+2) + (1+2+3) + (1+2+3+4) + ...... + (1+2+3+4+...+n)
Last Updated :
25 Jul, 2025
Given a positive integer n, find the sum of the following series, where the i-th term is the sum of the first i natural numbers:
(1) + (1+2) + (1+2+3) + ... + (1+2+3+...+n)
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 the implementation of the above series:
C++
// 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;
}
// 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.
# 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))
// 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.
<script>
// JavaScript Program for Sum of the series
// Function to find sum of given series
function sumOfSeries(n)
{
let sum = 0;
for (let i = 1 ; i <= n ; i++)
for (let j = 1 ; j <= i ; j++)
sum += j;
return sum;
}
// Driver code
let n = 10;
document.write(sumOfSeries(n));
</script>
<?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.
?>
Output :
220
Time Complexity: O(n^2)
Space Complexity: O(1)
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++
// 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);
}
// 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.
# 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))
// 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.
<script>
// JavaScript program For Sum of the series
// Function to find sum of given series
function sumOfSeries(n)
{
return (n * (n + 1) *
(2 * n + 4)) / 12;
}
// Driver code
let n = 10;
document.write(sumOfSeries(n));
</script>
<?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.
?>
Output :
220
Time Complexity: O(1)
Auxiliary Space: O(1)
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