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 Nth 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,
second group has 4 terms,
.
.
.
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 above approach:

C++

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// 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;
}

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Java

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// 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

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Python3

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# 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))

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C#

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// 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

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PHP

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<?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
?>

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Output:

255


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