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Sum of series 1^2 + 3^2 + 5^2 + . . . + (2*n – 1)^2

Given a series 12 + 32 + 52 + 72 + . . . + (2*n – 1)2, find sum of the series.

Examples:

Input : n = 4
Output : 84
Explanation : 
sum = 12 + 32 + 52 + 72
    = 1 + 9 + 25 + 49
    = 84

Input : n = 10 
Output : 1330
Explanation :
sum = 12 + 32 + 52 + 72 + 92 + 112 + 132 + 152 + 172 + 192
    = 1 + 9 + 24 + 49 + . . . + 361
    = 1330



C++

// Program to find sum of series
// 1^2 + 3^2 + 5^2 + . . . + (2*n - 1)^2.
#include <bits/stdc++.h>
using namespace std;
  
// Function to find sum of series.
int sumOfSeries(int n)
{
    int sum = 0;
    for (int i = 1; i <= n; i++)
        sum = sum + (2 * i - 1) * (2 * i - 1);
    return sum;
}
  
// Driver code
int main()
{
    int n = 10;
  
    cout << sumOfSeries(n);
  
    return 0;
}

Java

// Program to find sum of series
// 1^2 + 3^2 + 5^2 + . . . + (2*n - 1)^2.
  
import java.io.*;
  
class GFG {
     
// Function to find sum of series.
 static int sumOfSeries(int n)
{
    int sum = 0;
    for (int i = 1; i <= n; i++)
       sum = sum + (2 * i - 1) * (2 * i - 1);
    return sum;
}
  
// Driver code
  public static void  main(String[] args)
{
    int n = 10;
    System.out.println( sumOfSeries(n));    
}
      
}

Python3

# Python Program to find sum of series
# 1^2 + 3^2 + 5^2 + . . . + (2*n - 1)^2.
  
import math
  
# Function to find sum of series.
def sumOfSeries(n):
  
    sum = 0
    for i in range(1,n+1):
        sum = sum + (2 * i - 1) * (2 * i - 1)
    return sum
      
# driver code
n= 10
print(sumOfSeries(n))
  
# This code is contributed by Gitanjali.

C#

// C# Program to find sum of series
// 1^2 + 3^2 + 5^2 + . . . + (2*n - 1)^2.
using System;
  
class GFG {
      
// Function to find sum of series.
static int sumOfSeries(int n)
{
    int sum = 0;
    for (int i = 1; i <= n; i++)
        sum = sum + (2 * i - 1) * (2 * i - 1);
      
    return sum;
}
  
// Driver code
public static void Main()
{
    int n = 10;
    Console.Write( sumOfSeries(n)); 
}
}
  
/* This code is contributed by vt_m*/

/div>

PHP

<?php
// PHP Program to find sum of series
// 1^2 + 3^2 + 5^2 + . . . + (2*n - 1)^2.
  
// Function to find sum of series.
function sumOfSeries($n)
{
    $sum = 0;
    for ($i = 1; $i <= $n; $i++)
        $sum = $sum + (2 * $i - 1) * 
                      (2 * $i - 1);
    return $sum;
}
  
// Driver code
$n = 10;
echo(sumOfSeries($n));
  
// This code is contributed by Ajit.
?>

Output:

1330

Time Complexity : O(n)

Another approach : Using formula to find sum of series :

    12 + 32 + 52 + 
     72 + . . . + (2*n - 1)2 
      = (n * (2 * n - 1) * (2 * n + 1)) / 3.


Please refer sum of squares of even and odd numbers for proof.

C++

// Program to find sum of series
// 1^2 + 3^2 + 5^2 + . . . + (2*n - 1)^2.
#include <bits/stdc++.h>
using namespace std;
  
// Function that find sum of series.
int sumOfSeries(int n)
{
    // Formula to find sum of series.
    return (n * (2 * n - 1) * (2 * n + 1)) / 3;
}
  
// Driver code
int main()
{
    int n = 10;
    cout << sumOfSeries(n);
    return 0;
}

Java

// Java Program to find sum of series
// 1^2 + 3^2 + 5^2 + . . . + (2*n - 1)^2.
  
import java.io.*;
import java.util.*;
  
class GFG {
      
// Function to find sum of series.
static int sumOfSeries(int n)
{
   // Formula to find sum of series.
    return (n * (2 * n - 1) * (2 * n + 1)) / 3;
  
}
  
// Driver function
   public static void main (String[] args) {
   int n=10;
    System.out.println(sumOfSeries(n)); 
      
}
  
}
  
// This code is contributed by Gitanjali.

Python3

# Python Program to find sum of series
# 1^2 + 3^2 + 5^2 + . . . + (2*n - 1)^2.
  
import math
  
# Function to find sum of series.
def sumOfSeries(n):
  
   # Formula to find sum of series.
    return int((n * (2 * n - 1) * (2 * n + 1)) / 3)
   
# driver code
n=10
print(sumOfSeries(n))
  
# This code is contributed by Gitanjali.

C#

// C# Program to find sum of series
// 1^2 + 3^2 + 5^2 + . . . + (2*n - 1)^2.
using System;
  
class GFG {
      
// Function to find sum of series.
static int sumOfSeries(int n)
{
    // Formula to find sum of series.
    return (n * (2 * n - 1) * (2 * n + 1)) / 3;
}
  
// Driver function
public static void Main () 
{
    int n = 10;
    Console.Write(sumOfSeries(n)); 
}
}
  
// This code is contributed by vt_m.

PHP

<?php
// PHP Program to find sum of series
// 1^2 + 3^2 + 5^2 + . . . + (2*n - 1)^2.
  
// Function that find sum of series.
function sumOfSeries($n)
{
    // Formula to find sum of series.
    return ($n * (2 * $n - 1) * 
                 (2 * $n + 1)) / 3;
}
  
// Driver code
$n = 10;
echo(sumOfSeries($n));
  
// This code is contributed by Ajit.
?>

Output:

1330

Time Complexity: O(1)



This article is attributed to GeeksforGeeks.org

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