Program to calculate sum of an Infinite Arithmetic-Geometric Sequence

Given three integers A, D, and R representing the first term, common difference, and common ratio of an infinite Arithmetic-Geometric Progression, the task is to find the sum of the given infinite Arithmetic-Geometric Progression such that the absolute value of R is always less than 1.

Examples:

Input: A = 0, D = 1, R = 0.5
Output: 0.666667

Input: A = 2, D = 3, R = -0.3
Output: 0.549451

Approach: An arithmetic-geometric sequence is the result of the term-by-term multiplication of the geometric progression series with the corresponding terms of an arithmetic progression series. The series is given by:

a, (a + d) * r, (a + 2 * d) * r², (a + 3 * d) * r³, …, [a + (N ? 1) * d] * r^{(N ? 1)}.

The N^th term of the Arithmetic-Geometric Progression is given by:

=> 

The sum of the Arithmetic-Geometric Progression is given by:

=>

where, |r| < 1.

Below is the implementation of the above approach:

0.666667

Time Complexity: O(1)
Auxiliary Space: O(1),  since no extra space has been taken.