Find three number in G.P. whose sum is 52 and the sum of whose products in pairs is 624

Answer

To find three numbers in Geometric Progression (G.P.) whose sum is 52 and the sum of their products in pairs is 624, we can follow these steps:

### Step 1: Define the three numbers in G.P.
Let the three numbers be:
- \( \frac{a}{r} \) (first term)
- \( a \) (second term)
- \( ar \) (third term)

### Step 2: Write the equations based on the given conditions
From the problem, we have two conditions:
1. The sum of the numbers:
\[ \frac{a}{r} + a + ar = 52 \]
This can be simplified to:
\[ a\left(\frac{1}{r} + 1 + r\right) = 52 \quad \text{(Equation 1)} \]

2. The sum of the products of the numbers taken two at a time:
\[ \left(\frac{a}{r} \cdot a\right) + \left(a \cdot ar\right) + \left(ar \cdot \frac{a}{r}\right) = 624 \]
This simplifies to:
\[ a^2\left(\frac{1}{r} + r + 1\right) = 624 \quad \text{(Equation 2)} \]

### Step 3: Relate the two equations
From Equation 1, we can express \( a \):
\[ a = \frac{52}{\frac{1}{r} + 1 + r} \]
Substituting this expression for \( a \) into Equation 2 gives:
\[ \left(\frac{52}{\frac{1}{r} + 1 + r}\right)^2 \left(\frac{1}{r} + r + 1\right) = 624 \]
This simplifies to:
\[ \frac{2704}{\left(\frac{1}{r} + 1 + r\right)} = 624 \]
Cross-multiplying gives:
\[ 2704 = 624 \left(\frac{1}{r} + 1 + r\right) \]
Thus:
\[ \frac{1}{r} + 1 + r = \frac{2704}{624} = \frac{169}{39} \]

### Step 4: Solve for \( r \)
Multiplying through by \( r \) gives:
\[ 1 + r + r^2 = \frac{169r}{39} \]
Rearranging leads to:
\[ 39r^2 - 130r + 39 = 0 \]

### Step 5: Use the quadratic formula
Using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 39 \), \( b = -130 \), and \( c = 39 \).
- The discriminant \( D = (-130)^2 - 4 \cdot 39 \cdot 39 = 16900 - 6084 = 10716 \).

Calculating \( r \):
\[ r = \frac{130 \pm \sqrt{10716}}{78} \]
Calculating \( \sqrt{10716} \approx 103.5 \):
\[ r \approx \frac{130 \pm 103.5}{78} \]
This gives two possible values for \( r \).

### Step 6: Calculate the values of \( a \) and the three numbers
Using the values of \( r \) obtained, substitute back to find \( a \) and thus the three numbers in G.P.

### Final Answer
The three numbers in G.P. are \( 4, 12, 36 \) or \( 36, 12, 4 \) depending on the value of \( r \).

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