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Sum of N-terms of geometric progression for larger values of N | Set 2 (Using recursion)

  • Difficulty Level : Medium
  • Last Updated : 22 Jun, 2021

A Geometric series is a series with a constant ratio between successive terms. The first term of the series is denoted by a and the common ratio is denoted by r. The series looks like this:- a, ar, ar^2, ar^3, ar^4,...
The task is to find the sum of such a series, mod M.

Examples:  

Input:  a = 1, r = 2, N = 10000, M = 10000
Output:  8751

Input:  a = 1, r = 4, N = 10000, M = 100000
Output:  12501 

Approach: 

  1. To find the sum of series a + ar + ar^2 + ar^3 + . . . . + ar^N   we can easily take a as common and find the sum of 1 + r + r^2 + r^3 + . . . . + r^N   and multiply it with a.
  2. Steps to find the sum of the above series. 
    • Here, it can be resolved that: 
      [1 + r + r^2 + r^3 + . . . + r^(2*n+1)] = (1+r)*(1 + (r^2) + (r^2)^2 + (r^2)^3 + . . . + (r^2)^n)

If we denote, 
Sum(r, n) = 1 + r + r^2 + r^3 + . . . . + r^N.   then, 
Sum(r, 2 * n + 1) = (1 + r) * Sum(r^2, n).   and, 
Sum(r, 2 * n) = 1 + (r * (1 + r) * Sum(r^2, n - 1)).
This will work as our recursive case. 

  • So, the base cases are:
Sum(r, 0) = 1.
Sum(r, 1) = 1 + r.

Below is the implementation of the above approach.  



C++




// C++ implementation to
// illustrate the program
#include <iostream>
using namespace std;
 
// Function to calculate the sum
// recursively
int SumGPUtil(long long int r,
              long long int n,
              long long int m)
{
     
    // Base cases
    if (n == 0)
        return 1;
    if (n == 1)
        return (1 + r) % m;
     
    long long int ans;
    // If n is odd
    if (n % 2 == 1)
    {
        ans = (1 + r) *
              SumGPUtil((r * r) % m,
                        (n - 1) / 2, m);
    }
    else
    {
         
        // If n is even
        ans = 1 + (r * (1 + r) *
             SumGPUtil((r * r) % m,
                       (n / 2) - 1, m));
    }
    return (ans % m);
}
 
// Function to print the value of Sum
void SumGP(long long int a,
           long long int r,
           long long int N,
           long long int M)
{
    long long int answer;
     
    answer = a * SumGPUtil(r, N, M);
    answer = answer % M;
     
    cout << answer << endl;
}
 
// Driver Code
int main()
{
     
    // First element
    long long int a = 1;
     
    // Common difference
    long long int r = 4;
     
    // Number of elements
    long long int N = 10000;
     
    // Mod value
    long long int M = 100000;
 
    SumGP(a, r, N, M);
 
    return 0;
}
 
// This code is contributed by sanjoy_62

Java




// Java implementation to
// illustrate the program
import java.io.*;
 
class GFG{
 
// Function to calculate the sum
// recursively
static long SumGPUtil(long r, long n,
                      long m)
{
     
    // Base cases
    if (n == 0)
        return 1;
    if (n == 1)
        return (1 + r) % m;
     
    long ans;
     
    // If n is odd
    if (n % 2 == 1)
    {
        ans = (1 + r) *
              SumGPUtil((r * r) % m,
                        (n - 1) / 2, m);
    }
    else
    {
        // If n is even
        ans = 1 + (r * (1 + r) *
             SumGPUtil((r * r) % m,
                       (n / 2) - 1, m));
    }
     
    return (ans % m);
}
 
// Function to prlong the value of Sum
static void SumGP(long a, long r,
                  long N, long M)
{
    long answer;
    answer = a * SumGPUtil(r, N, M);
    answer = answer % M;
     
    System.out.println(answer);
}
 
// Driver Code
public static void main (String[] args)
{
     
    // First element
    long a = 1;
     
    // Common difference
    long r = 4;
     
    // Number of elements
    long N = 10000;
     
    // Mod value
    long M = 100000;
 
    SumGP(a, r, N, M);
}
}
 
// This code is contributed by sanjoy_62

Python3




# Python3 implementation to illustrate the program
 
# Function to calculate the sum
# recursively
def SumGPUtil (r, n, m):
     
    # Base cases
    if n == 0: return 1
    if n == 1: return (1 + r) % m
   
    # If n is odd
    if n % 2 == 1:
        ans = (1 + r) * SumGPUtil(r * r % m,
                                  (n - 1)//2,
                                  m)
    else:
        #If n is even
        ans = 1 + r * (1 + r) * SumGPUtil(r * r % m,
                                          n//2 - 1,
                                          m)
   
    return ans % m
 
# Function to print the value of Sum
def SumGP (a, r, N, M):
     
    answer = a * SumGPUtil(r, N, M)
    answer = answer % M
    print(answer)
 
#Driver Program
if __name__== '__main__':
 
    a = 1 # first element
    r = 4 # common difference
    N = 10000 # Number of elements
    M = 100000 # Mod value
 
    SumGP(a, r, N, M)

C#




// C# implementation to
// illustrate the program
using System;
 
class GFG{
 
// Function to calculate the sum
// recursively
static long SumGPUtil(long r, long n,
                      long m)
{
     
    // Base cases
    if (n == 0)
        return 1;
    if (n == 1)
        return (1 + r) % m;
     
    long ans;
     
    // If n is odd
    if (n % 2 == 1)
    {
        ans = (1 + r) *
              SumGPUtil((r * r) % m,
                        (n - 1) / 2, m);
    }
    else
    {
         
        // If n is even
        ans = 1 + (r * (1 + r) *
             SumGPUtil((r * r) % m,
                       (n / 2) - 1, m));
    }
    return (ans % m);
}
 
// Function to prlong the value of Sum
static void SumGP(long a, long r,
                  long N, long M)
{
    long answer;
    answer = a * SumGPUtil(r, N, M);
    answer = answer % M;
     
    Console.WriteLine(answer);
}
 
// Driver Code
public static void Main()
{
     
    // First element
    long a = 1;
     
    // Common difference
    long r = 4;
     
    // Number of elements
    long N = 10000;
     
    // Mod value
    long M = 100000;
 
    SumGP(a, r, N, M);
}
}
 
// This code is contributed by sanjoy_62

Javascript




<script>
 
// Javascript implementation to
// illustrate the program
 
// Function to calculate the sum
// recursively
function SumGPUtil(r, n, m)
{
       
    // Base cases
    if (n == 0)
        return 1;
    if (n == 1)
        return (1 + r) % m;
       
    let ans;
       
    // If n is odd
    if (n % 2 == 1)
    {
        ans = (1 + r) *
              SumGPUtil((r * r) % m,
                        (n - 1) / 2, m);
    }
    else
    {
        // If n is even
        ans = 1 + (r * (1 + r) *
             SumGPUtil((r * r) % m,
                       (n / 2) - 1, m));
    }
       
    return (ans % m);
}
   
// Function to prlet the value of Sum
function SumGP(a, r, N, M)
{
    let answer;
    answer = a * SumGPUtil(r, N, M);
    answer = answer % M;
       
    document.write(answer);
}
   
 
// Driver Code
     
    // First element
    let a = 1;
       
    // Common difference
    let r = 4;
       
    // Number of elements
    let N = 10000;
       
    // Mod value
    let M = 100000;
   
    SumGP(a, r, N, M);
                       
</script>
Output: 
12501

 

Time complexity: O(log N)
 

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