Sum of all subsequences of an array

Last Updated : 18 Sep, 2023

Given an array of n integers. Find the sum of all possible subsequences of an array.

Examples :

Input : arr[] = { 1, 2 }
Output : 6
All possible subsequences are {}, {1}, {2} 
and { 1, 2 }

Input : arr[] = { 1, 2, 3 }
Output : 24

We have already discussed two different solutions in below post.
Sum of all Subarrays | Set 1

In this post a different solution is discussed. Let us take a closer look at the problem and try to find a pattern

Let a[] = { 1, 2, 3 }

All subsequences are {}, {1}, {2}, {3}, {1, 2}, 
                     {1, 3}, {2, 3}, {1, 2, 3}

So sum of subsequences are 0 + 1 + 2 + 3 + 3 + 
                             4 + 5 + 6 = 24

Here we can observe that in sum every elements 
occurs 4 times. Or in general every element 
will occur 2^(n-1) times. And we can also 
observe that sum of array elements is 6. So
final result will be 6*4.

In general we can find sum of all subsequences by adding all elements of array multiplied by 2^(n-1) where n is number of elements in array.

Implementation:

C++

// CPP program to find sum of 
// all subarrays of array 
#include <bits/stdc++.h> 
using namespace std; 
  
// To find sum of all subsequences 
int findSum(int arr[], int n) 
{ 
    // Sum all array elements 
    int sum = 0; 
    for (int i = 0; i < n; i++) 
        sum += arr[i]; 
  
    // Result is sum * 2^(n-1) 
    return sum * (1 << (n - 1)); 
} 
  
// Driver program to test findSum() 
int main() 
{ 
    int arr[] = { 1, 2 }; 
    int n = sizeof(arr) / sizeof(arr[0]); 
    cout << findSum(arr, n); 
    return 0; 
} 

Java

// Java program to find sum of 
// all subarrays of array 
  
public class Main { 
    // To find sum of all subsequences 
    static int findSum(int arr[], int n) 
    { 
        // Sum all array elements 
        int sum = 0; 
        for (int i = 0; i < n; i++) 
            sum += arr[i]; 
  
        // Result is sum * 2^(n-1) 
        return sum * (1 << (n - 1)); 
    } 
  
    // Driver program to test findSum() 
    public static void main(String[] args) 
    { 
        int arr[] = { 1, 2 }; 
        int n = arr.length; 
        System.out.print(findSum(arr, n)); 
    } 
} 

Python

# Python program to find sum of 
# all subarrays of array 
  
# To find sum of all subsequences 
def findSum(arr, n): 
      
    # Sum all array elements 
    sum = 0
    for i in range(n): 
        sum += arr[i] 
   
    # Result is sum * 2^(n-1) 
    return sum * (1 << (n - 1)) 
   
# Driver program to test findSum() 
arr = [1, 2] 
n = len(arr) 
print findSum(arr, n) 
  
# This code is submitted by Sachin Bisht 

C#

// C# program to find sum of  
// all subarrays of array  
using System; 
  
class GFG 
{ 
  
// To find sum of all subsequences 
static int findSum(int []arr, int n)  
{  
    // Sum all array elements  
    int sum = 0;  
    for (int i = 0; i < n; i++)  
        sum += arr[i];  
  
    // Result is sum * 2^(n-1)  
    return sum * (1 << (n - 1));  
}  
  
// Driver Code 
static public void Main () 
{ 
    int []arr = { 1, 2 };  
    int n = arr.Length;  
    Console.WriteLine(findSum(arr, n));  
}  
}  
  
// This code is contributed by ajit 

PHP

<?php 
// PHP program to find sum of 
// all subarrays of array 
  
// To find sum of all subsequences 
  
function findSum($arr, $n) 
{ 
    // Sum all array elements 
    $sum = 0; 
    for ($i = 0; $i < $n; $i++) 
        $sum += $arr[$i]; 
  
    // Result is sum * 2^(n-1) 
    return $sum * (1 << ($n - 1)); 
} 
  
// Driver Code 
$arr = array( 1, 2 ); 
$n = sizeof($arr); 
  
echo findSum($arr, $n); 
  
// This code is contributed by ajit 
?> 

Javascript

<script> 
  
// Javascript program to find sum of  
// all subarrays of array  
  
// To find sum of all subsequences 
function findSum(arr, n)  
{  
      
    // Sum all array elements  
    let sum = 0;  
    for(let i = 0; i < n; i++)  
        sum += arr[i];  
  
    // Result is sum * 2^(n-1)  
    return sum * (1 << (n - 1));  
}  
  
// Driver code 
let arr = [ 1, 2 ];  
let n = arr.length;  
  
document.write(findSum(arr, n));  
  
// This code is contributed by rameshtravel07 
  
</script>

Output

To check a number is palindrome or not without using any extra space

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Arrays

Sum of all subsequences of an array

C++

Java

Python

C#

PHP

Javascript

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