A geometric progression consists of 500 terms. Sum of the terms occupying the odd places is `P_1` and the sum of the terms occupying the even places is `P_2` . Find the common ratio.

Options

A. `P_2//P_1 `
B. `P_1//P_2`
C. `P_2 + P_1//P_1 `
D. `P_2 + P_1//P_2`

Answer

To solve the problem, we need to find the common ratio of a geometric progression (GP) that consists of 500 terms, given that the sum of the terms in the odd positions is \( P_1 \) and the sum of the terms in the even positions is \( P_2 \).

### Step-by-Step Solution:

1. **Understanding the Geometric Progression**:
Let the first term of the GP be \( a \) and the common ratio be \( r \). The terms of the GP can be expressed as:
- 1st term: \( a \)
- 2nd term: \( ar \)
- 3rd term: \( ar^2 \)
- 4th term: \( ar^3 \)
- ...
- 500th term: \( ar^{499} \)

2. **Identifying Odd and Even Position Terms**:
- The terms in the odd positions are: \( a, ar^2, ar^4, \ldots \) (up to the 500th term)
- The terms in the even positions are: \( ar, ar^3, ar^5, \ldots \) (up to the 500th term)

3. **Counting the Terms**:
Since there are 500 terms in total:
- The number of odd-positioned terms is \( 250 \) (1st, 3rd, ..., 499th).
- The number of even-positioned terms is \( 250 \) (2nd, 4th, ..., 500th).

4. **Calculating \( P_1 \) (Sum of Odd Position Terms)**:
The sum of the first \( n \) terms of a GP is given by the formula:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
For the odd-positioned terms:
\[ P_1 = a \left(1 + r^2 + r^4 + \ldots + r^{498}\right) = a \frac{1 - r^{500}}{1 - r^2} \]

5. **Calculating \( P_2 \) (Sum of Even Position Terms)**:
For the even-positioned terms:
\[ P_2 = ar \left(1 + r^2 + r^4 + \ldots + r^{498}\right) = ar \frac{1 - r^{500}}{1 - r^2} \]

6. **Finding the Ratio \( \frac{P_2}{P_1} \)**:
Now, we can find the ratio of \( P_2 \) to \( P_1 \):
\[ \frac{P_2}{P_1} = \frac{ar \frac{1 - r^{500}}{1 - r^2}}{a \frac{1 - r^{500}}{1 - r^2}} = r \]

7. **Conclusion**:
Therefore, the common ratio \( r \) can be expressed as:
\[ r = \frac{P_2}{P_1} \]

### Final Answer:
The common ratio \( r \) is \( \frac{P_2}{P_1} \).