Geometric Progression

1. Understand the problem:

We are given a geometric progression (G.P.) with first term 'a' and common ratio 'r'. We need to find the ratio of the sum of the first n terms ( ) to the sum of the first 2n terms ( ).

2. Recall the formula for sum of a G.P.:

The sum of the first n terms of a G.P. is given by:

Similarly, the sum of the first 2n terms is:

3. Compute the ratio :

Divide by :

4. Simplify the expression:

Notice that can be factored as . Thus:

5. Match the result to the options:

The simplified form corresponds to option (B) .

Correct Answer: (B)