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Sum of elements of a Geometric Progression (GP) in a given range

  • Last Updated : 06 Apr, 2021

Given a Geometric Progression series in arr[] and Q queries in the form of [L, R], where L is the left boundary of the range and R is the right boundary. The task is to find the sum of the Geometric Progression elements in the given range.

Note: The range is 1-indexed and1 ≤ L, R ≤ N, where N is the size of arr.
Examples: 

Input: arr[] = {2, 4, 8, 16, 32, 64, 128, 256}, Q = [[2, 4], [2, 6], [5, 8]] 
Output: 
28 
124 
480
Explanation: 
Range 1: arr = {4, 8, 16}. Therefore sum = 28 
Range 2: arr = {4, 8, 16, 32, 64}. Therefore sum = 124 
Range 3: arr = {32, 64, 128, 256}. Therefore sum = 480

Input: arr[] = {7, 7, 7, 7, 7, 7}, Q = [[1, 6], [2, 4], [3, 3]] 
Output: 
42
21 

Explanation: 
Range 1: arr = {7, 7, 7, 7, 7, 7}. Therefore sum = 42
Range 2: arr = {7, 7, 7}. Therefore sum = 21 
Range 3: arr = {7}. Therefore sum = 7

Approach: Since the given sequence is an Geometric progression, the sum can be easily found out in two steps efficiently:

  1. Get the first element of the range.
  2. If d = 1, then multiply d*k to it, else multiply the (dk – 1)/(d – 1) to it, where d is the common ratio of the GP and k is number of elements in the range.

For example: 
Suppose a[i] be the first element of the range, d be the common ratio of GP and k be the number of elements in the given range. 
Then the sum of the range would be 

= a[i] + a[i+1] + a[i+2] + ….. + a[i+k-1] 
= a[i] + (a[i] * d) + (a[i] * d * d) + …. + (a[i] *  dk) 
= a[i] *  (1 + d + … + dk
= a[i] * (dk – 1)/(d – 1)

Below is the implementation of the above approach: 

C++




// C++ program to find the sum
// of elements of an GP in the
// given range
#include <bits/stdc++.h>
using namespace std;
 
// Function to find sum in the given range
int findSum(int arr[], int n,
            int left, int right)
{
     
    // Find the value of k
    int k = right - left + 1;
 
    // Find the common difference
    int d = arr[1] / arr[0];
 
    // Find the sum
    int ans = arr[left - 1];
     
    if (d == 1)
        ans = ans * d * k;
    else
        ans = ans * ((int)pow(d, k) - 1 /
                                 (d - 1));
         
    return ans;
}
 
// Driver Code
int main()
{
    int arr[] = { 2, 4, 8, 16, 32,
                  64, 128, 256 };
    int queries = 3;
    int q[][2] = { { 2, 4 }, { 2, 6 },
                   { 5, 8 } };
     
    int n = sizeof(arr) / sizeof(arr[0]);
     
    for(int i = 0; i < queries; i++)
        cout << (findSum(arr, n, q[i][0], q[i][1]))
             << endl;
 
    return 0;
}
 
// This code is contributed by divyeshrabadiya07


Java




// Java program to find the sum
// of elements of an GP in the
// given range
import java.io.*;
import java.util.*;
 
class GFG{
     
// Function to find sum in the given range
static int findSum(int[] arr, int n,
                int left, int right)
{
     
    // Find the value of k
    int k = right - left + 1;
 
    // Find the common difference
    int d = arr[1] / arr[0];
 
    // Find the sum
    int ans = arr[left - 1];
     
    if (d == 1)
        ans = ans * d * k;
    else
        ans = ans * ((int)Math.pow(d, k) - 1 /
                                (d - 1));
         
    return ans;
}
 
// Driver Code
public static void main(String args[])
{
    int[] arr = { 2, 4, 8, 16, 32,
                64, 128, 256 };
    int queries = 3;
    int[][] q = { { 2, 4 }, { 2, 6 }, { 5, 8 } };
     
    int n = arr.length;
     
    for(int i = 0; i < queries; i++)
        System.out.println(findSum(arr, n, q[i][0],
                                        q[i][1]));
}
}
 
// This code is contributed by offbeat


Python3




# Python3 program to
# find the sum of elements
# of an GP in the given range
 
# Function to find sum in the given range
def findSum(arr, n, left, right):
 
    # Find the value of k
    k = right - left + 1
 
    # Find the common difference
    d = arr[1] // arr[0]
 
    # Find the sum
    ans = arr[left - 1]
    if d == 1:
        ans = ans * d * k
    else:
        ans = ans * (d ** k - 1) // (d -1)
    return ans
 
# Driver code
if __name__ == '__main__':
    arr = [ 2, 4, 8, 16, 32, 64, 128, 256 ]
    queries = 3
    q = [[ 2, 4 ], [ 2, 6 ], [ 5, 8 ]]
    n = len(arr)
 
    for i in range(queries):
        print(findSum(arr, n, q[i][0], q[i][1]))


C#




// C# program to find the sum
// of elements of an GP in the
// given range
using System;
 
class GFG{
     
// Function to find sum in the given range
static int findSum(int[] arr, int n,
                   int left, int right)
{
     
    // Find the value of k
    int k = right - left + 1;
 
    // Find the common difference
    int d = arr[1] / arr[0];
 
    // Find the sum
    int ans = arr[left - 1];
     
    if (d == 1)
        ans = ans * d * k;
    else
        ans = ans * ((int)Math.Pow(d, k) - 1 /
                                      (d - 1));
         
    return ans;
}
 
// Driver Code
public static void Main(string []args)
{
    int[] arr = { 2, 4, 8, 16, 32,
                  64, 128, 256 };
                   
    int queries = 3;
    int[,] q = { { 2, 4 }, { 2, 6 }, { 5, 8 } };
     
    int n = arr.Length;
     
    for(int i = 0; i < queries; i++)
        Console.Write(findSum(arr, n, q[i, 0],
                                      q[i, 1]) + "\n");
}
}
 
// This code is contributed by rutvik_56


Javascript




<script>
// JavaScript program to find the sum
// of elements of an GP in the
// given range
 
// Function to find sum in the given range
function findSum(arr, n, left, right) {
 
    // Find the value of k
    let k = right - left + 1;
 
    // Find the common difference
    let d = arr[1] / arr[0];
 
    // Find the sum
    let ans = arr[left - 1];
 
    if (d == 1)
        ans = ans * d * k;
    else
        ans = ans * (Math.pow(d, k) - 1 / (d - 1));
 
    return ans;
}
 
// Driver Code
let arr = [2, 4, 8, 16, 32,
    64, 128, 256];
 
let queries = 3;
 
let q = [[2, 4], [2, 6],
[5, 8]];
 
let n = arr.length;
 
for (let i = 0; i < queries; i++)
    document.write(findSum(arr, n, q[i][0], q[i][1]));
 
// This code is contributed by blalverma92
 
</script>


Output: 

28
124
480

 

  • Time complexity: O(Q) 
  • Space complexity: O(1)


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