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 spliting 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++

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// 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;
}

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Java

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// 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

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Python

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# 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

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C#

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//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..

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PHP

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<?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..
?>

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Output:

Sum = 34


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