If `(a)/(b) = (b)/(c ) = (c )/(d)` , then `(a)/(d)` in terms of `b` and `c` is equal to :

Options

A. `((b)/(c ))^(3)`
B. `((b)/(c ))^(2)`
C. `((b)/(c ))^(1.5)`
D. `((b)/(c ))^(0.5)`

Answer

To solve the problem step by step, we start with the given relationships:

1. **Given Relationships**:
\[ \frac{a}{b} = \frac{b}{c} = \frac{c}{d} = k \]
where \( k \) is some constant.

2. **Expressing Variables in Terms of \( k \)**:
From the first equation, we can express \( a \) in terms of \( b \):
\[ a = kb \]
From the second equation, we can express \( b \) in terms of \( c \):
\[ b = kc \]
From the third equation, we can express \( c \) in terms of \( d \):
\[ c = kd \]

3. **Finding \( a \) in terms of \( c \) and \( d \)**:
Now, we can express \( a \) in terms of \( c \) and \( d \):
- Substitute \( b \) in the equation for \( a \):
\[ a = k(kc) = k^2c \]
- Substitute \( c \) in the equation for \( a \):
\[ a = k^2(kd) = k^3d \]

4. **Finding \( \frac{a}{d} \)**:
Now we can find \( \frac{a}{d} \):
\[ \frac{a}{d} = \frac{k^3d}{d} = k^3 \]

5. **Expressing \( k \) in terms of \( b \) and \( c \)**:
We know from our earlier equations:
- From \( \frac{b}{c} = k \), we have:
\[ k = \frac{b}{c} \]

6. **Substituting \( k \) back into \( \frac{a}{d} \)**:
Therefore, substituting \( k \) back into our expression for \( \frac{a}{d} \):
\[ \frac{a}{d} = \left(\frac{b}{c}\right)^3 \]

Thus, the final answer is:
\[ \frac{a}{d} = \frac{b^3}{c^3} \]