Find nth term of a given recurrence relation

Let a_n be a sequence of numbers, which is defined by the recurrence relation a₁=1 and a_n+1/a_n=2ⁿ. The task is to find the value of log₂(a_n) for a given n.

Examples:

Input: 5
Output: 10
Explanation: 
log2(an) = (n * (n - 1)) / 2
= (5*(5-1))/2
= 10

Input: 100
Output: 4950

, 

We multiply all of the above in order to reach


Since   .

Then, 

Substituting n+1 for n: 

So, log_{2}(a_{n})=\frac{n(n-1)}{2}      

Below is the implementation of the above approach as follows: 

C++

// C++ program to find nth term of
// a given recurrence relation

#include <bits/stdc++.h>

using namespace std;

// function to return required value
int sum(int n)
{

    // Get the answer
    int ans = (n * (n - 1)) / 2;

    // Return the answer
    return ans;
}

// Driver program
int main()
{

    // Get the value of n
    int n = 5;

    // function call to print result
    cout << sum(n);

    return 0;
}

Java

// Java program to find nth term
// of a given recurrence relation
import java.util.*;

class solution
{
static int sum(int n)
{
    // Get the answer
    int ans = (n * (n - 1)) / 2;

    // Return the answer
    return ans;
}

// Driver code
public static void main(String arr[])
{

    // Get the value of n
    int n = 5;

    // function call to print result
    System.out.println(sum(n));
}
}
//This code is contributed byte
//Surendra_Gangwar

Python3

# Python3 program to find nth
# term of a given recurrence
# relation

# function to return
# required value
def sum(n):

    # Get the answer
    ans = (n * (n - 1)) / 2;

    # Return the answer
    return ans

# Driver Code

# Get the value of n
n = 5

# function call to prresult
print(int(sum(n)))

# This code is contributed by Raj

C#

// C# program to find nth term
// of a given recurrence relation
using System;

class GFG
{
static int sum(int n)
{
    // Get the answer
    int ans = (n * (n - 1)) / 2;

    // Return the answer
    return ans;
}

// Driver code
public static void Main()
{

    // Get the value of n
    int n = 5;

    // function call to print result
    Console.WriteLine(sum(n));
}
}

// This code is contributed byte
// inder_verma

PHP

<?php
// PHP program to find nth term of
// a given recurrence relation

// function to return required value
function sum($n)
{

    // Get the answer
    $ans = ($n * ($n - 1)) / 2;

    // Return the answer
    return $ans;
}

// Driver Code

// Get the value of n
$n = 5;

// function call to print result
echo sum($n);

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

Javascript

<script>
// Javascript program to find nth term of
// a given recurrence relation

// function to return required value
function sum(n)
{

    // Get the answer
    let ans = parseInt((n * (n - 1)) / 2);

    // Return the answer
    return ans;
}

// Driver program

// Get the value of n
let n = 5;

// function call to print result
document.write(sum(n));

// This code is contributed by subham348.
</script>

Output:

10

Time Complexity: O(1)

Auxiliary Space: O(1) // Because using constant variables