Find the sum of the three numbers in G.P. whose products is 216 and the sum of the products if taken in pairs is 156 :

Options

A. A. 28
B. B. 21
C. C. `35//4`
D. D. none of these

Answer

To solve the problem of finding the sum of three numbers in a geometric progression (G.P.) whose product is 216 and the sum of the products taken in pairs is 156, we can follow these steps:

### Step 1: Define the Numbers
Let the three numbers in G.P. be:
- \( a/r \)
- \( a \)
- \( ar \)

### Step 2: Set Up the Equations
From the problem, we have two conditions:
1. The product of the numbers:
\[ \left(\frac{a}{r}\right) \cdot a \cdot (ar) = 216 \]
Simplifying this gives:
\[ \frac{a^3}{r} = 216 \quad \text{(Equation 1)} \]

2. The sum of the products taken in pairs:
\[ \left(\frac{a}{r}\right) \cdot a + a \cdot (ar) + \left(\frac{a}{r}\right) \cdot (ar) = 156 \]
Simplifying this gives:
\[ \frac{a^2}{r} + a^2 + a^2 = 156 \]
Which can be rewritten as:
\[ \frac{a^2}{r} + 2a^2 = 156 \quad \text{(Equation 2)} \]

### Step 3: Solve for \( a \)
From Equation 1, we can express \( a^3 \):
\[ a^3 = 216r \]

### Step 4: Substitute \( a^2 \) in Equation 2
From Equation 1, we can find \( a^2 \):
\[ a^2 = \sqrt[3]{(216r)^2} = 36r^{2/3} \]

Now substitute \( a^2 \) into Equation 2:
\[ \frac{36r^{2/3}}{r} + 2(36r^{2/3}) = 156 \]
This simplifies to:
\[ 36r^{-1/3} + 72r^{2/3} = 156 \]

### Step 5: Multiply through by \( r^{1/3} \)
To eliminate the fraction, multiply through by \( r^{1/3} \):
\[ 36 + 72r = 156r^{1/3} \]

### Step 6: Rearrange the Equation
Rearranging gives:
\[ 156r^{1/3} - 72r - 36 = 0 \]

### Step 7: Solve for \( r \)
Let \( x = r^{1/3} \), then \( r = x^3 \):
\[ 156x - 72x^3 - 36 = 0 \]
This can be rearranged to:
\[ 72x^3 - 156x + 36 = 0 \]

### Step 8: Factor or Use the Quadratic Formula
This cubic equation can be solved using numerical methods or factoring. After finding the roots, we can find the corresponding values of \( a \) and \( r \).

### Step 9: Find the Numbers
Once we have \( a \) and \( r \), we can find the three numbers:
- \( a/r \)
- \( a \)
- \( ar \)

### Step 10: Calculate the Sum
Finally, we can find the sum of the three numbers:
\[ \text{Sum} = \frac{a}{r} + a + ar \]