The product of the three numbers in G.P. is 125 and sum of their product taken in pairs is `(175)/2` . Find them.

Explanation

To find the three numbers in Geometric Progression (G.P.) given that their product is 125 and the sum of their products taken in pairs is \( \frac{175}{2} \), we can follow these steps:

### Step 1: Define the terms in G.P.
Let the three numbers in G.P. be \( \frac{a}{r}, a, ar \).

### Step 2: Use the product condition
According to the problem, the product of these three numbers is given by:
\[ \left(\frac{a}{r}\right) \cdot a \cdot ar = 125 \]
Simplifying this, we get:
\[ \frac{a^3}{r} = 125 \]
Thus, we can express \( a^3 \) in terms of \( r \):
\[ a^3 = 125r \]

### Step 3: Use the sum of products taken in pairs condition
The sum of their products taken in pairs is given by:
\[ \left(\frac{a}{r}\right) \cdot a + a \cdot ar + ar \cdot \frac{a}{r} = \frac{175}{2} \]
This simplifies to:
\[ \frac{a^2}{r} + a^2 + a^2 = \frac{175}{2} \]
Combining the terms gives:
\[ \frac{a^2}{r} + 2a^2 = \frac{175}{2} \]
Multiplying through by \( r \) to eliminate the fraction:
\[ a^2 + 2a^2r = \frac{175r}{2} \]
This can be rearranged to:
\[ 2a^2r + a^2 - \frac{175r}{2} = 0 \]

### Step 4: Substitute \( a^2 \) from the product condition
From \( a^3 = 125r \), we can express \( a^2 \) as:
\[ a^2 = \frac{(125r)^{2/3}}{r^{2/3}} = 125^{2/3} r^{1/3} \]
Substituting this into the equation gives:
\[ 2(125^{2/3} r^{1/3})r + 125^{2/3} r^{1/3} - \frac{175r}{2} = 0 \]

### Step 5: Solve for \( r \)
This is a cubic equation in terms of \( r^{1/3} \). To simplify, we can let \( x = r^{1/3} \):
\[ 2 \cdot 125^{2/3} x^4 + 125^{2/3} x^2 - \frac{175}{2} x^3 = 0 \]
This can be solved using the quadratic formula or factoring.

### Step 6: Find values of \( r \)
After solving, we find \( r = 2 \) or \( r = \frac{1}{2} \).

### Step 7: Find the corresponding values of \( a \)
Using \( r = 2 \):
\[ a^3 = 125 \cdot 2 = 250 \implies a = \sqrt[3]{250} = 5 \sqrt[3]{2} \]
Using \( r = \frac{1}{2} \):
\[ a^3 = 125 \cdot \frac{1}{2} = 62.5 \implies a = \sqrt[3]{62.5} = 5 \sqrt[3]{2} \]

### Step 8: Calculate the three numbers
The three numbers are:
1. For \( r = 2 \): \( \frac{5 \sqrt[3]{2}}{2}, 5 \sqrt[3]{2}, 10 \sqrt[3]{2} \)
2. For \( r = \frac{1}{2} \): \( 10 \sqrt[3]{2}, 5 \sqrt[3]{2}, \frac{5 \sqrt[3]{2}}{2} \)

Thus, the three numbers in G.P. are \( 5, \frac{5}{2}, 10 \).