If a,b,c,d,e,f, g are in contiued proportion, then the value of a.b.c.e.f.g is

Options

A. `d^(3)`
B. `d^(7)`
C. `d^(6)`
D. None of these

Answer

To solve the problem, we need to understand the concept of continuous proportion. If \( a, b, c, d, e, f, g \) are in continuous proportion, it means that the ratios of consecutive terms are equal. This can be expressed mathematically as:

\[ \frac{a}{b} = \frac{b}{c} = \frac{c}{d} = \frac{d}{e} = \frac{e}{f} = \frac{f}{g} \]

From this, we can derive that:

1. \( a \cdot d = b^2 \)
2. \( b \cdot e = c^2 \)
3. \( c \cdot f = d^2 \)
4. \( d \cdot g = e^2 \)

Now, we want to find the value of \( a \cdot b \cdot c \cdot e \cdot f \cdot g \).

### Step-by-Step Solution:

1. **Express all variables in terms of one variable**:
Let's express all variables in terms of \( a \). We can use the relationships derived from the continuous proportion.

- Let \( b = k \cdot a \)
- Then \( c = k \cdot b = k^2 \cdot a \)
- Continuing this, we find:
- \( d = k^3 \cdot a \)
- \( e = k^4 \cdot a \)
- \( f = k^5 \cdot a \)
- \( g = k^6 \cdot a \)

2. **Substitute these expressions into the product**:
Now we substitute these expressions into \( a \cdot b \cdot c \cdot e \cdot f \cdot g \):

\[ a \cdot b \cdot c \cdot e \cdot f \cdot g = a \cdot (k \cdot a) \cdot (k^2 \cdot a) \cdot (k^4 \cdot a) \cdot (k^5 \cdot a) \cdot (k^6 \cdot a) \]

3. **Combine the terms**:
This simplifies to:

\[ = a^6 \cdot k^{1+2+4+5+6} = a^6 \cdot k^{18} \]

4. **Conclusion**:
Since \( k \) is a constant ratio, the value of \( a \cdot b \cdot c \cdot e \cdot f \cdot g \) is proportional to \( a^6 \) multiplied by some constant \( k^{18} \).

Thus, the final answer is:

\[ a \cdot b \cdot c \cdot e \cdot f \cdot g = a^6 \cdot k^{18} \]