The series obtained by adding the terms of a G.P. is called the geometric series. Let be the sum of the first terms of the G.P. The term is , the term is , etc. Hence
Multiplying both members of (1) by , we get
On subtracting each side of (2) from the corresponding side of (1), we obtain
If the common ratio in a G.P. is more than , then each successive term is greater than the previous one and the sum of the terms grows very rapidly and tends to infinity as tends to infinity.
If the common ratio is less than , then each term will be smaller than the previous one and the total will increase but will not exceed a finite value as tends to infinity.
Using G.P., find the sum of .
Substituting these values in the given formula, we get
Given , , , find .