If a, b, c be in G.P., then the expression `a^(2)b^(2)c^(2) ((1)/(a^(3)) + (1)/(b^(3)) + (1)/(c^(3)))=`

Options

A. a + b + c
B. `a^(3) + b^(3) + c^(3)`
C. ab + bc + ca
D. none

Answer

To solve the problem, we need to evaluate the expression \( a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) \) given that \( a, b, c \) are in geometric progression (G.P.).

### Step-by-Step Solution:

1. **Understanding G.P.**: Since \( a, b, c \) are in G.P., we can express \( b \) in terms of \( a \) and \( c \). Specifically, we have:
\[ b^2 = ac \quad \text{(1)} \]

2. **Rewrite the Expression**: The expression we need to evaluate is:
\[ a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) \]
We can rewrite the sum inside the parentheses:
\[ \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} = \frac{b^3 c^3 + a^3 c^3 + a^3 b^3}{a^3 b^3 c^3} \]

3. **Finding a Common Denominator**: The common denominator for the fractions is \( a^3 b^3 c^3 \). Thus, we can express the sum as:
\[ \frac{b^3 c^3 + a^3 c^3 + a^3 b^3}{a^3 b^3 c^3} \]

4. **Substituting the Expression**: Now, substituting this back into our expression gives:
\[ a^2 b^2 c^2 \cdot \frac{b^3 c^3 + a^3 c^3 + a^3 b^3}{a^3 b^3 c^3} \]

5. **Simplifying the Expression**: The \( a^2 b^2 c^2 \) in the numerator will cancel with part of the denominator:
\[ = \frac{a^2 b^2 c^2 (b^3 c^3 + a^3 c^3 + a^3 b^3)}{a^3 b^3 c^3} = \frac{(b^3 c^3 + a^3 c^3 + a^3 b^3)}{a b c} \]

6. **Using Equation (1)**: From equation (1), we know \( b^2 = ac \). Therefore, we can express \( a \) and \( c \) in terms of \( b \):
\[ a = \frac{b^2}{c}, \quad c = \frac{b^2}{a} \]

7. **Final Expression**: After substituting and simplifying, we can conclude:
\[ = a^3 + b^3 + c^3 \]

8. **Conclusion**: The final value of the expression is:
\[ a^3 + b^3 + c^3 \]

### Final Answer:
The expression evaluates to \( a^3 + b^3 + c^3 \).