Sum of the series 1 + (1+3) + (1+3+5) + (1+3+5+7) + …… + (1+3+5+7+…+(2n-1))

Given a positive integer n. The problem is to find the sum of the given series 1 + (1+2) + (1+2+3) + (1+2+3+4) + …… + (1+2+3+4+…+n), where i-th term in the series is the sum of first i odd natural numbers.

Examples:

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

Input : n = 5
Output : 55
(1) + (1+3) + (1+3+5) + (1+3+5+7) + (1+3+5+7+9) = 55



Naive Approach: Using two loops get the sum of each i-th term and then add those sum to the final sum.

C++

filter_none

edit
close

play_arrow

link
brightness_4
code

// C++ implementation to find the 
// sum of the given series
#include <bits/stdc++.h>
  
using namespace std;
  
// functionn to find the 
// sum of the given series
int sumOfTheSeries(int n)
{
    int sum = 0;
    for (int i = 1; i <= n; i++) {
  
        // first term of each i-th term
        int k = 1;
        for (int j = 1; j <= i; j++) {
            sum += k;
  
            // next term
            k += 2;
        }
    }
  
    // required sum
    return sum;
}
  
// Driver program 
int main()
{
    int n = 5;
    cout << "Sum = "
         << sumOfTheSeries(n);
    return 0;
}

chevron_right


Java

filter_none

edit
close

play_arrow

link
brightness_4
code

// Java implementation to find 
// the sum of the given series
import java.util.*;
  
class GFG {
      
    // functionn to find the sum
    // of the given series
    static int sumOfTheSeries(int n)
    {
        int sum = 0;
        for (int i = 1; i <= n; i++)
        {
       
            // first term of each 
            // i-th term
            int k = 1;
            for (int j = 1; j <= i; j++)
            {
                sum += k;
       
                // next term
                k += 2;
            }
        }
       
        // required sum
        return sum;
    }
  
    /* Driver program */
    public static void main(String[] args) 
    {
         int n = 5;
         System.out.println("Sum = "
                        sumOfTheSeries(n));
    }
}
  
// This code is contributed by Arnav Kr. Mandal.

chevron_right


Python3

filter_none

edit
close

play_arrow

link
brightness_4
code

# Python3 implementation to find
# the sum of the given series
  
# functionn to find the sum
# of the given series
def sumOfTheSeries( n ):
    sum = 0
    for i in range(1, n + 1):
          
        # first term of each i-th term
        k = 1
        for j in range(1,i+1):
            sum += k
              
            # next term
            k += 2
              
    # required sum
    return sum
      
# Driver program
n = 5
print("Sum =", sumOfTheSeries(n))
  
# This code is contributed by "Sharad_Bhardwaj".

chevron_right


C#

filter_none

edit
close

play_arrow

link
brightness_4
code

// C# implementation to find
// the sum of the given series
using System;
  
class GFG {
  
    // functionn to find the sum
    // of the given series
    static int sumOfTheSeries(int n)
    {
        int sum = 0;
        for (int i = 1; i <= n; i++) {
  
            // first term of each
            // i-th term
            int k = 1;
            for (int j = 1; j <= i; j++) {
                sum += k;
  
                // next term
                k += 2;
            }
        }
  
        // required sum
        return sum;
    }
  
    /* Driver program */
    public static void Main()
    {
        int n = 5;
        Console.Write("Sum = "
                     sumOfTheSeries(n));
    }
}
  
// This code is contributed by vt_m.

chevron_right


php

filter_none

edit
close

play_arrow

link
brightness_4
code

<?php
  
// php implementation to find the 
// sum of the given series
  
// functionn to find the 
// sum of the given series
function sumOfTheSeries($n)
{
    $sum = 0;
    for ($i = 1; $i <= $n; $i++) {
   
        // first term of each i-th term
        $k = 1;
        for ($j = 1; $j <= $i; $j++) {
            $sum += $k;
   
            // next term
            $k += 2;
        }
    }
   
    // required sum
    return $sum;
}
   
// Driver program 
    $n = 5;
    echo "Sum = "
         . sumOfTheSeries($n);
  
// This code is contributed by Sam007
?>

chevron_right



Output:

Sum = 55

Efficient Approach:
Let an be the n-th term of the given series.

an = (1 + 3 + 5 + 7 + (2n-1))
   = sum of first n odd numbers
   = n2

Refer this post for the proof of above formula.

Now,

Refer this post for the proof of above formula.

C++

filter_none

edit
close

play_arrow

link
brightness_4
code

// C++ implementation to find the sum
// of the given series
#include <bits/stdc++.h>
using namespace std;
  
// functionn to find the sum
// of the given series
int sumOfTheSeries(int n)
{
    // required sum
    return (n * (n + 1) / 2) *
            (2 * n + 1) / 3;
}
  
// Driver program to test above
int main()
{
    int n = 5;
    cout << "Sum = "
         << sumOfTheSeries(n);
    return 0;
}

chevron_right


Java

filter_none

edit
close

play_arrow

link
brightness_4
code

// Java implementation to find 
// the sum of the given series
import java.io.*;
  
class GfG {
      
// function to find the sum
// of the given series
static int sumOfTheSeries(int n)
{
    // required sum
    return (n * (n + 1) / 2) *
            (2 * n + 1) / 3;
}
      
  
// Driver program to test above
public static void main (String[] args) 
{
    int n = 5;
      
    System.out.println("Sum = "
                sumOfTheSeries(n));
  
}
  
}
  
// This code is contributed by Gitanjali.

chevron_right


Python3

filter_none

edit
close

play_arrow

link
brightness_4
code

# Python3 implementation to find
# the sum of the given series
  
# functionn to find the sum
# of the given series
def sumOfTheSeries( n ):
      
    # required sum
    return int((n * (n + 1) / 2) *
            (2 * n + 1) / 3)
              
# Driver program to test above
n = 5
print("Sum =", sumOfTheSeries(n))
  
# This code is contributed by "Sharad_Bhardwaj".

chevron_right


C#

filter_none

edit
close

play_arrow

link
brightness_4
code

// C# implementation to find
// the sum of the given series
using System;
  
class GfG {
  
    // function to find the sum
    // of the given series
    static int sumOfTheSeries(int n)
    {
        // required sum
        return (n * (n + 1) / 2) * 
                      (2 * n + 1) / 3;
    }
  
    // Driver program to test above
    public static void Main()
    {
        int n = 5;
  
        Console.Write("Sum = "
                   sumOfTheSeries(n));
    }
}
  
// This code is contributed by vt_m.

chevron_right


PHP

filter_none

edit
close

play_arrow

link
brightness_4
code

<?php
// PHP implementation to find the sum
// of the given series
  
// functionn to find the sum
// of the given series
function sumOfTheSeries($n)
{
      
    // required sum
    return ($n * ($n + 1) / 2) *
              (2 * $n + 1) / 3;
}
  
    // Driver Code
    $n = 5;
    echo "Sum = "
        . sumOfTheSeries($n);
          
// This code is contributed by Sam007
?>

chevron_right



Output:

Sum = 55


My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.



Improved By : Sam007