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Program to find the Nth term of the series 3, 7, 13, 21, 31…..
  • Difficulty Level : Hard
  • Last Updated : 25 Feb, 2021

Given a number N, the task is to find the Nth term of this series:
 

3, 7, 13, 21, 31, …….

Examples: 
 

Input: N = 4
Output: 21
Explanation:
Nth term = (pow(N, 2) + N + 1)
         = (pow(4, 2) + 4 + 1)
         = 21

Input: N = 11
Output: 133

 

Approach:
Sum = 0+3+7+13+21+31+........+a_{n-1} + a_n\\ Sum = 3+7+13+21+31+...+a_{n-2}+a_{n-1}+a_n
Subtracting these two equations we get 
$ 0=3+\left \{\frac{n-1}{2}\right \}[2*4 + (n-2)*2]-a_n\\ =3+\left \{\frac{n-1}{2}\right \}[8 + 2n-4]-a_n\\ =3+\left \{\frac{n-1}{2}\right \}[2n+4]-a_n\\ a_n=3+(n-1)(n+2)\\ a_n=n^2+n+1 $
Therefore, the Nth Term of the given series is: 
 



a_n=n^2+n+1

Below is the implementation of the above approach: 
 

C++




// CPP program to find the Nth term of given series.
#include <iostream>
#include <math.h>
using namespace std;
 
// Function to calculate sum
long long int getNthTerm(long long int N)
{
    // Return Nth term
    return (pow(N, 2) + N + 1);
}
 
// driver code
int main()
{
    // declaration of number of terms
    long long int N = 11;
 
    // Get the Nth term
    cout << getNthTerm(N);
 
    return 0;
}


Java




// Java code to find the Nth term of given series.
import java.util.*;
 
class solution
{
 
// Function to calculate sum
static long getNthTerm(long N)
{
     
   // Return Nth term
    return ((int)Math.pow(N, 2) + N + 1);
}
 
//Driver program
public static void main(String arr[])
{
     
   // declaration of number of terms
    long N = 11;
 
    // Get the Nth term
    System.out.println(getNthTerm(N));
 
}
}
//THis code is contibuted by
//Surendra_Gangwar


Python3




# Python3 Code to find the
# Nth term of given series.
 
# Function to calculate sum
def getNthTerm(N):
     
    # Return Nth term
    return (pow(N, 2) + N + 1)
 
# driver code
if __name__=='__main__':
     
# declaration of number of terms
    N = 11
     
# Get the Nth term
    print(getNthTerm(N))
 
# This code is contributed by
# Sanjit_Prasad


C#




// C# code to find the Nth
// term of given series.
using System;
 
class GFG
{
 
// Function to calculate sum
static long getNthTerm(long N)
{
     
// Return Nth term
    return ((int)Math.Pow(N, 2) + N + 1);
}
 
// Driver Code
static public void Main ()
{
     
    // declaration of number
    // of terms
    long N = 11;
 
    // Get the Nth term
    Console.Write(getNthTerm(N));
}
}
 
// This code is contibuted by Raj


PHP




<?php
// PHP program to find the
// Nth term of given series
 
// Function to calculate sum
function getNthTerm($N)
{
    // Return Nth term
    return (pow($N, 2) + $N + 1);
}
 
// Driver code
 
// declaration of number of terms
$N = 11;
 
// Get the Nth term
echo getNthTerm($N);
 
// This code is contributed by Raj
?>


Javascript




<script>
// JavaScript program to find the Nth term of given series.
 
// Function to calculate sum
function getNthTerm(N)
{
    // Return Nth term
    return (Math.pow(N, 2) + N + 1);
}
   
// driver code
 
   // declaration of number of terms
   let N = 11;
   
   // Get the Nth term
   document.write(getNthTerm(N));
   
// This code is contributed by Surbhi Tyagi
 
</script>


Output: 

133

 

Time Complexity: O(1)
 

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