Sum of first n term of Series 3, 5, 9, 17, 33….

Given n, we need to find sum of first n terms of the series represented as Sn = 3 + 5 + 9 + 17 + 33 … upto n
Examples:

Input : 2
Output : 8
3 + 5 = 8

Input : 5
Output : 67
3 + 5 + 9 + 17 + 33 = 67

Let, the nth term be denoted by tn. 
This problem can easily be solved by splitting each term as follows :

Sn = 3 + 5 + 9 + 17 + 33…… 
Sn = (2+1) + (4+1) + (8+1) + (16+1) +…… 
Sn = (2+1) + (2*2+1) + (2*2*2+1) + (2*2*2*2+1) +……+ ((2*2*2..unto n times) + 1)

We observed that the nth term can be written in terms of powers of 2 and 1. 
Hence, the sum of first n terms is given as follows:

Sn = (2+1) + (4+1) + (8+1) + (16+1) +……+ upto n terms 
Sn = (1 + 1 + 1 + 1 + …unto n terms) + (2 + 4 + 8 + 16 + …upto nth power of 2) 
In above formula, 
2 + 4 + 8 + 16…. is a G.P. 
It’s sum of first n terms is given by 2*(2^n-1)/(2-1) = 2^(n+1) – 2 (using G.P. formula) 
Sn = n + 2*(2^n – 1) 
Sn = 2^(n+1) + n -2

C++

// C++ program to find sum of first n terms
#include <bits/stdc++.h>
using namespace std;

int calculateSum(int n)
{
    // Sn = n*(4*n*n + 6*n - 1)/3
    return (pow(2, n + 1) + n - 2);
}

// Driver code
int main()
{
    // number of terms to be included in sum
    int n = 4;

    // find the Sn
    cout << "Sum = " << calculateSum(n);

    return 0;
}

Java

// Java program to find
// sum of first n terms
import java.util.*;

class GFG
{
static int calculateSum(int n)
{
    // Sn = n*(4*n*n + 6*n - 1)/3
    return ((int)Math.pow(2, n + 1) +
                             n - 2);
}

// Driver Code
public static void main(String args[])
{
    // number of terms to
    // be included in sum
    int n = 4;

    // find the Sn
    System.out.println("Sum = " +
                calculateSum(n));
}
}

// This code is contributed
// by Kirti_Mangal

Python

# Python program to find sum
# of n terms of the series
def calculateSum(n):

    return (2**(n + 1) + n - 2)

# Driver Code

# number of terms for the sum
n = 4

# find the Sn
print("Sum =", calculateSum(n))

# This code is contributed
# by Surendra_Gangwar

C#

//C# program to find
// sum of first n terms
using System;

class GFG
{
static int calculateSum(int n)
{
    // Sn = n*(4*n*n + 6*n - 1)/3
    return ((int)Math.Pow(2, n + 1) +
                            n - 2);
}

// Driver Code
public static void Main()
{
    // number of terms to
    // be included in sum
    int n = 4;

    // find the Sn
    Console.WriteLine("Sum = " +
                calculateSum(n));
}
}

// This code is contributed
// by inder_verma..

PHP

<?php
// PHP program to find sum
// of first n terms
function calculateSum( $n)
{
    // Sn = n*(4*n*n + 6*n - 1)/3
    return (pow(2, $n + 1) + $n - 2);
}

// Driver code

// number of terms to be
// included in sum
$n = 4;

// find the Sn
echo "Sum = " , calculateSum($n);

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

Javascript

<script>
// Java script program to find
// sum of first n terms

function calculateSum( n)
{
    // Sn = n*(4*n*n + 6*n - 1)/3
    return (Math.pow(2, n + 1) +
                             n - 2);
}

// Driver Code

    // number of terms to
    // be included in sum
    let n = 4;

    // find the Sn
    document.write("Sum = " +
                calculateSum(n));

// This code is contributed by mohan pavan

</script>

Output:

Sum = 34