Let `a_(n)` be the nth term of a G.P. of positive real numbers. If `overset(200)underset(n=1)Sigma a_(2n)=4 and overset(200)underset(n=1)Sigma a_(2n-1)=5`, then the common ratio of the G.P. is

Answer

To find the common ratio of the geometric progression (G.P.) given the conditions of the problem, we can follow these steps:

### Step 1: Understand the terms of the G.P.
Let the first term of the G.P. be \( a \) and the common ratio be \( r \). The \( n \)-th term of the G.P. can be expressed as:
\[ a_n = a \cdot r^{n-1} \]

### Step 2: Write the sums for even and odd indexed terms
The sum of the even indexed terms from \( n=1 \) to \( n=200 \) is:
\[ \sum_{n=1}^{200} a_{2n} = a \cdot r^{1} + a \cdot r^{3} + a \cdot r^{5} + \ldots + a \cdot r^{399} \]
This can be factored as:
\[ = a \cdot (r + r^3 + r^5 + \ldots + r^{399}) \]

The sum of the odd indexed terms from \( n=1 \) to \( n=200 \) is:
\[ \sum_{n=1}^{200} a_{2n-1} = a + a \cdot r^{2} + a \cdot r^{4} + \ldots + a \cdot r^{398} \]
This can be factored as:
\[ = a \cdot (1 + r^{2} + r^{4} + \ldots + r^{398}) \]

### Step 3: Use the formula for the sum of a geometric series
The sum of a geometric series can be calculated using the formula:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]

For the even indexed terms:
\[ \sum_{n=1}^{200} a_{2n} = a \cdot \frac{r(1 - r^{400})}{1 - r^2} \]
For the odd indexed terms:
\[ \sum_{n=1}^{200} a_{2n-1} = a \cdot \frac{1(1 - r^{400})}{1 - r^2} \]

### Step 4: Set up the equations based on the given sums
From the problem, we know:
\[ \sum_{n=1}^{200} a_{2n} = 4 \quad \text{and} \quad \sum_{n=1}^{200} a_{2n-1} = 5 \]

Thus, we have:
1. \( a \cdot \frac{r(1 - r^{400})}{1 - r^2} = 4 \) (Equation 1)
2. \( a \cdot \frac{(1 - r^{400})}{1 - r^2} = 5 \) (Equation 2)

### Step 5: Divide Equation 1 by Equation 2
Dividing Equation 1 by Equation 2 gives:
\[ \frac{a \cdot \frac{r(1 - r^{400})}{1 - r^2}}{a \cdot \frac{(1 - r^{400})}{1 - r^2}} = \frac{4}{5} \]
This simplifies to:
\[ r = \frac{4}{5} \]

### Step 6: Conclusion
Thus, the common ratio \( r \) of the G.P. is:
\[ \boxed{0.8} \]