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Find sum of N-th group of Natural Numbers

  • Last Updated : 10 May, 2021

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, 
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++




// 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




// 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

Python3




# 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#




// 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
// 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
?>

Javascript




<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>
Output: 
255

 


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