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

  • Last Updated : 20 Apr, 2021

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

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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*/

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.
?>

Javascript




<script>
 
// JavaScript 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)
{
    let sum = 0;
    for(let i = 1; i <= n; i++)
       sum = sum + (2 * i - 1) *
                   (2 * i - 1);
        
    return sum;
}
 
// Driver Code
let n = 10;
 
document.write(sumOfSeries(n)); 
   
// This code is contributed by avijitmondal1998
 
</script>
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.
?>

Javascript




<script>
// Javascript 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
let n = 10;
document.write(sumOfSeries(n));
 
// This code is contributed by _saurabh_jaiswal.
 
</script>
Output: 
1330

 

Time Complexity: O(1)
 




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