Sum of Nterms of geometric progression for larger values of N  Set 2 (Using recursion)
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:
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:
 To find the sum of series we can easily take a as common and find the sum of and multiply it with a.

Steps to find the sum of above series.

Here, it can be resolved that:
. 
If we denote,
then,
and,
This will work as our recursive case.

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

Here, it can be resolved that:
Below is the implementation of the above approach.
# 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 diffrence N = 10000 # Number of elements M = 100000 # Mod value SumGP(a, r, N, M) 
12501
Time complexity: O(log N)
Recommended Posts:
 Geometric Progression
 Program to print GP (Geometric Progression)
 Number of GP (Geometric Progression) subsequences of size 3
 Program for Nth term of Geometric Progression series
 Removing a number from array to make it Geometric Progression
 Minimum number of operations to convert a given sequence into a Geometric Progression
 Check whether nodes of Binary Tree form Arithmetic, Geometric or Harmonic Progression
 Find geometric sum of the series using recursion
 Find larger of x^y and y^x
 Check if a larger number divisible by 36
 Find the larger exponential among two exponentials
 Smallest triangular number larger than p
 Number of Larger Elements on right side in a string
 Geometric Median
 Geometric mean (Two Methods)
 Find N Geometric Means between A and B
 Program for sum of geometric series
 Sum of Arithmetic Geometric Sequence
 Arithmetic Progression
 Harmonic Progression
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.