The sum of 3 numbers in G.P.is 38 and their product is 1728 find the greatest number:

Options

A. 24
B. 18
C. 16
D. 8

Answer

To solve the problem, we need to find three numbers in geometric progression (G.P.) given that their sum is 38 and their product is 1728. Let's denote the three numbers as \( \frac{a}{r}, a, ar \).

### Step 1: Set up the equations
From the problem, we have two equations:
1. The sum of the numbers:
\[ \frac{a}{r} + a + ar = 38 \]
2. The product of the numbers:
\[ \frac{a}{r} \cdot a \cdot ar = 1728 \]

### Step 2: Simplify the product equation
The product can be simplified as follows:
\[ \frac{a^3}{r} = 1728 \]
Multiplying both sides by \( r \):
\[ a^3 = 1728r \]

### Step 3: Substitute \( a^3 \) into the sum equation
Now, we can express \( a \) in terms of \( r \). From \( a^3 = 1728r \), we can find \( a \):
\[ a = \sqrt[3]{1728r} \]
Now we can substitute \( a \) back into the sum equation:
\[ \frac{\sqrt[3]{1728r}}{r} + \sqrt[3]{1728r} + \sqrt[3]{1728r} \cdot r = 38 \]

### Step 4: Solve for \( a \)
To simplify, let's first find \( \sqrt[3]{1728} \):
\[ \sqrt[3]{1728} = 12 \]
Thus, we can write:
\[ \frac{12\sqrt[3]{r}}{r} + 12\sqrt[3]{r} + 12\sqrt[3]{r} \cdot r = 38 \]
This simplifies to:
\[ \frac{12}{\sqrt[3]{r^2}} + 12\sqrt[3]{r} + 12r\sqrt[3]{r} = 38 \]

### Step 5: Rearranging the equation
Let \( x = \sqrt[3]{r} \). Then \( r = x^3 \), and we can rewrite the equation:
\[ \frac{12}{x^2} + 12x + 12x^4 = 38 \]
Multiply through by \( x^2 \) to eliminate the fraction:
\[ 12 + 12x^3 + 12x^6 = 38x^2 \]
Rearranging gives:
\[ 12x^6 + 12x^3 - 38x^2 + 12 = 0 \]

### Step 6: Solve the polynomial equation
This is a polynomial equation in \( x \). We can use numerical methods or factorization to find the roots. For simplicity, we can use the Rational Root Theorem or synthetic division to find rational roots.

### Step 7: Find the values of \( r \)
Assuming we find \( r = \frac{3}{2} \) and \( r = \frac{2}{3} \) as potential roots, we can substitute these back to find \( a \).

### Step 8: Calculate the three numbers
Using \( r = \frac{3}{2} \):
1. First number: \( \frac{12}{\frac{3}{2}} = 12 \cdot \frac{2}{3} = 8 \)
2. Second number: \( a = 12 \)
3. Third number: \( ar = 12 \cdot \frac{3}{2} = 18 \)

### Step 9: Identify the greatest number
The three numbers are \( 8, 12, 18 \). The greatest number is:
\[ \text{Greatest number} = 18 \]

### Final Answer
The greatest number is **18**.

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