Find sum of N-th group of Natural Numbers

Given a series of natural numbers divided into groups as: (1, 2), (3, 4, 5, 6), (7, 8, 9, 10, 11, 12), (13, 14, 15, 16, 17, 18, 19, 20)..... and so on. Given a number N, the task is to find the sum of the numbers in the N^th group.

Examples: 

Input : N = 3
Output : 57
Numbers in 3rd group are:
7, 8, 9, 10, 11, 12

Input : N = 10 
Output : 2010 

The first group has 2 terms, 
the second group has 4 terms, 
. 
. 
. 
the nth group has 2n terms.
Now, 
The last term of the first group is 2 = 1 × (1 + 1)
The last term of the second group is 6 = 2 × (2 + 1)
The last term of the third group is 12 = 3 × (3 + 1)
The last term of the fourth group is 20 = 4 × (4 + 1) 
. 
. 
. 
The last term of the nth group = n(n + 1).
Therefore, the sum of the numbers in the nth group is:  

= sum of all the numbers upto nth group - sum of all the numbers upto (n - 1)th group
= [1 + 2 +........+ n(n + 1)] - [1 + 2 +........+ (n - 1 )((n - 1) + 1)]
= \frac{n(n+1)[n(n+1)+1]}{2} - \frac {n(n-1)[n(n-1)+1]}{2}   
= \frac{n[(n+1)(n(n+1)+1)-(n-1)(n(n-1)+1)]}{2}   
= \frac{n[n[(n+1)^{2} - (n-1)^{2}]+2]}{2}   
= n(2n^{2}+1)  

Below is the implementation of the above approach: 
 

// C++ program to find sum in Nth group
#include<bits/stdc++.h>
using namespace std; 

//calculate sum of Nth group
int nth_group(int n){
     return n * (2 * pow(n, 2) + 1);
}

//Driver code
int main()
{

  int N = 5;
  cout<<nth_group(N);
  
  return 0;
}

// Java program to find sum
// in Nth group
import java.util.*;

class GFG
{

// calculate sum of Nth group
static int nth_group(int n)
{
    return n * (2 * (int)Math.pow(n, 2) + 1);
}

// Driver code
public static void main(String arr[])
{
    int N = 5;
    System.out.println(nth_group(N));
}
}

// This code is contributed by Surendra

# Python program to find sum in Nth group

# calculate sum of Nth group
def nth_group(n):
    return n * (2 * pow(n, 2) + 1)

# Driver code
N = 5
print(nth_group(N))

// C# program to find sum in Nth group

using System; 

class gfg
{
 //calculate sum of Nth group
 public static double nth_group(int n)
 {
    return n * (2 * Math.Pow(n, 2) + 1);
 }

 //Driver code
 public static int Main()
 {
   int N = 5;
   Console.WriteLine(nth_group(N));
   return 0;
 }
}
// This code is contributed by Soumik

<?php
// PHP program to find sum
// in Nth group

// calculate sum of Nth group
function nth_group($n)
{
    return $n * (2 * pow($n, 2) + 1);
}

// Driver code
$N = 5;
echo nth_group($N);

// This code is contributed 
// by jit_t
?>

<script>
    // Javascript program to find sum in Nth group
    
    //calculate sum of Nth group
    function nth_group(n)
    {
      return n * (2 * Math.pow(n, 2) + 1);
    }
    
   let N = 5;
   document.write(nth_group(N));
    
</script>

255

Time Complexity: O(1)

Auxiliary Space: O(1)