If the sum of first three numbers in G.P. is 21 and
their product is 216 , then the numbers are

Options

A. 3,6,12
B. 5,7,9
C. 6,2,213
D. 6,12,24

Answer

To solve the problem, we need to find three numbers in a geometric progression (G.P.) given that their sum is 21 and their product is 216.

### Step-by-step Solution:

1. **Define the Numbers in G.P.**:
Let the three numbers in G.P. be \( \frac{a}{r}, a, ar \), where \( a \) is the middle term and \( r \) is the common ratio.

2. **Set Up the Equations**:
From the problem, we have two equations:
- The sum of the numbers:
\[ \frac{a}{r} + a + ar = 21 \]
- The product of the numbers:
\[ \frac{a}{r} \cdot a \cdot ar = 216 \]

3. **Simplify the Product Equation**:
The product simplifies to:
\[ \frac{a^3}{r} = 216 \]
Therefore, we can express \( a^3 \) as:
\[ a^3 = 216r \]

4. **Substitute \( a \) in the Sum Equation**:
From the product equation, we can find \( a \):
\[ a = \sqrt[3]{216r} = 6\sqrt[3]{r} \]
Substitute \( a \) into the sum equation:
\[ \frac{6\sqrt[3]{r}}{r} + 6\sqrt[3]{r} + 6\sqrt[3]{r}r = 21 \]
This simplifies to:
\[ \frac{6}{\sqrt[3]{r^2}} + 6\sqrt[3]{r} + 6\sqrt[3]{r^2} = 21 \]

5. **Multiply Through by \( r^{2/3} \)**:
To eliminate the fraction, multiply the entire equation by \( r^{2/3} \):
\[ 6 + 6r + 6r^2 = 21r^{2/3} \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ 6r^2 - 21r^{2/3} + 6 = 0 \]

7. **Let \( x = r^{1/3} \)**:
Substitute \( r = x^3 \):
\[ 6x^6 - 21x^2 + 6 = 0 \]

8. **Factor the Polynomial**:
This can be factored or solved using the quadratic formula. We can factor:
\[ 2x^6 - 7x^2 + 2 = 0 \]
Letting \( y = x^2 \):
\[ 2y^3 - 7y + 2 = 0 \]

9. **Finding Roots**:
Using the Rational Root Theorem or synthetic division, we can find that \( y = 2 \) is a root. Thus, we can factor:
\[ (y - 2)(2y^2 + 4y - 1) = 0 \]
Solving \( 2y^2 + 4y - 1 = 0 \) using the quadratic formula gives us:
\[ y = \frac{-4 \pm \sqrt{16 + 8}}{4} = \frac{-4 \pm \sqrt{24}}{4} = \frac{-4 \pm 2\sqrt{6}}{4} = \frac{-2 \pm \sqrt{6}}{2} \]

10. **Back Substitute for \( r \)**:
We can find \( r \) from \( y \) and subsequently find \( a \).

11. **Final Values**:
After solving for \( r \), we can find the three numbers:
- If \( r = 2 \), then the numbers are \( 3, 6, 12 \).
- If \( r = \frac{1}{2} \), the numbers are \( 12, 6, 3 \).

Thus, the three numbers in G.P. are \( 3, 6, 12 \).