Sum of three consecutive terms in a G.P. is 42 their product is 512 . Find the largest of these numbers. :

Options

A. 28
B. 16
C. 32
D. none of these

Answer

To solve the problem, we need to find three consecutive terms in a geometric progression (G.P.) whose sum is 42 and product is 512. Let's denote the three terms as \( a \), \( ar \), and \( ar^2 \).

### Step-by-Step Solution:

1. **Set Up the Equations**:
- The sum of the three terms is given by:
\[ a + ar + ar^2 = 42 \]
- The product of the three terms is given by:
\[ a \cdot ar \cdot ar^2 = 512 \]
This simplifies to:
\[ a^3 r^3 = 512 \]

2. **Express the Product Equation**:
- Since \( 512 = 8^3 \), we can write:
\[ a^3 r^3 = 8^3 \]
- Taking the cube root of both sides gives:
\[ ar = 8 \]
- From this, we can express \( a \) in terms of \( r \):
\[ a = \frac{8}{r} \]

3. **Substitute \( a \) in the Sum Equation**:
- Substitute \( a = \frac{8}{r} \) into the sum equation:
\[ \frac{8}{r} + \frac{8}{r}r + \frac{8}{r}r^2 = 42 \]
- This simplifies to:
\[ \frac{8}{r} + 8 + 8r = 42 \]
- Multiply through by \( r \) to eliminate the fraction:
\[ 8 + 8r + 8r^2 = 42r \]

4. **Rearranging the Equation**:
- Rearranging gives:
\[ 8r^2 - 34r + 8 = 0 \]

5. **Using the Quadratic Formula**:
- We can solve this quadratic equation using the quadratic formula:
\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Here, \( a = 8 \), \( b = -34 \), and \( c = 8 \):
\[ r = \frac{34 \pm \sqrt{(-34)^2 - 4 \cdot 8 \cdot 8}}{2 \cdot 8} \]
\[ r = \frac{34 \pm \sqrt{1156 - 256}}{16} \]
\[ r = \frac{34 \pm \sqrt{900}}{16} \]
\[ r = \frac{34 \pm 30}{16} \]

6. **Finding the Values of \( r \)**:
- This gives us two possible values for \( r \):
\[ r = \frac{64}{16} = 4 \quad \text{and} \quad r = \frac{4}{16} = \frac{1}{4} \]

7. **Finding Corresponding Values of \( a \)**:
- For \( r = 4 \):
\[ a = \frac{8}{4} = 2 \]
The terms are \( 2, 8, 32 \).
- For \( r = \frac{1}{4} \):
\[ a = \frac{8}{\frac{1}{4}} = 32 \]
The terms are \( 32, 8, 2 \).

8. **Identifying the Largest Term**:
- In both cases, the largest term is \( 32 \).

### Final Answer:
The largest of the three consecutive terms in the G.P. is \( \boxed{32} \).