Find n-th term of series 3, 9, 21, 41, 71…

Given a mathematical series as 3, 9, 21, 41, 71… For a given integer n, you have to find the nth number of this series. 
Examples :

Input : n = 4 
Output : 41

Input : n = 2
Output : 9

Our first task for solving this problem is to crack the series. If you will gave a closer look on the series, for a general n-th term the value will be (?n²)+(?n)+1, where

• (?n²): is sum of squares of first n-natural numbers.
• (?n): is sum of first n-natural numbers.
• 1 is a simple constant value.

So, for calculating any n-th term of given series say f(n) we have: 
f(n) = (?n²)+(?n)+1
= ( ((n*(n+1)*(2n+1))/6) + (n*(n+1)/2) + 1 
= (n³+ 3n² + 2n + 3 ) /3

C++

// Program to calculate
// nth term of a series
#include <bits/stdc++.h>
using namespace std;

// func for calualtion
int seriesFunc(int n)
{
    // for summation of square
    // of first n-natural nos.
    int sumSquare = (n * (n + 1)
                  * (2 * n + 1)) / 6;

    // summation of first n natural nos.
    int sumNatural = (n * (n + 1) / 2);

    // return result
    return (sumSquare + sumNatural + 1);
}

// Driver Code
int main()
{
    int n = 8;
    cout << seriesFunc(n) << endl;

    n = 13;
    cout << seriesFunc(13);

    return 0;
}

Java

// Java Program to calculate
// nth term of a series
import java.io.*;

class GFG
{
    // func for calualtion
    static int seriesFunc(int n)
    {
        // for summation of square
        // of first n-natural nos.
        int sumSquare = (n * (n + 1)
                        * (2 * n + 1)) / 6;

        // summation of first n natural nos.
        int sumNatural = (n * (n + 1) / 2);

        // return result
        return (sumSquare + sumNatural + 1);
    }

    // Driver Code
    public static void main(String args[])
    {
        int n = 8;
        System.out.println(seriesFunc(n));

        n = 13;
        System.out.println(seriesFunc(13));
    }
}

// This code is contributed by Nikita Tiwari.

Python3

# Program to calculate
# nth term of a series

# func for calualtion
def seriesFunc(n):

    # for summation of square
    # of first n-natural nos.
    sumSquare = (n * (n + 1) *
                (2 * n + 1)) / 6

    # summation of first n
    # natural nos.
    sumNatural = (n * (n + 1) / 2)

    # return result
    return (sumSquare + sumNatural + 1)

# Driver Code
n = 8
print (int(seriesFunc(n)))

n = 13
print (int(seriesFunc(n)))

# This is code is contributed by Shreyanshi Arun.

C#

// C# program to calculate
// nth term of a series
using System;

class GFG
{
    // Function for calualtion
    static int seriesFunc(int n)
    {
        // For summation of square
        // of first n-natural nos.
        int sumSquare = (n * (n + 1)
                        * (2 * n + 1)) / 6;

        // summation of first n natural nos.
        int sumNatural = (n * (n + 1) / 2);

        // return result
        return (sumSquare + sumNatural + 1);
    }

    // Driver Code
    public static void Main()
    {
        int n = 8;
        Console.WriteLine(seriesFunc(n));

        n = 13;
        Console.WriteLine(seriesFunc(13));
    }
}

// This code is contributed by vt_m.

PHP

<?php
// Program to calculate
// nth term of a series

// func for calualtion
function seriesFunc($n)
{
    // for summation of square
    // of first n-natural nos.
    $sumSquare = ($n * ($n + 1)
                * (2 * $n + 1)) / 6;

    // summation of first n natural nos.
    $sumNatural = ($n * ($n + 1) / 2);

    // return result
    return ($sumSquare + $sumNatural + 1);
}

// Driver Code
$n = 8;
echo(seriesFunc($n) . "\n");

$n = 13;
echo(seriesFunc($n) . "\n");

// This code is contributed by Ajit.
?>

Javascript

<script>

// JavaScript Program to calculate
// nth term of a series

    // func for calualtion
    function seriesFunc(n)
    {
        // for summation of square
        // of first n-natural nos.
        let sumSquare = (n * (n + 1)
                        * (2 * n + 1)) / 6;

        // summation of first n natural nos.
        let sumNatural = (n * (n + 1) / 2);

        // return result
        return (sumSquare + sumNatural + 1);
    }

// Driver code

        let n = 8;
        document.write(seriesFunc(n) + "<br/>");

        n = 13;
        document.write(seriesFunc(13));

</script>

Output :

241
911

Time Complexity: O(1) since constant operations are performed

Auxiliary Space: O(1)