If `(a + be^(x))/(a-be^(x)) = (b + ce^(x))/(b-ce^(x)) = (c+de^(x))/(c - de^(x))` then a, b, c, d are in

Options

A. A.P.
B. G.P.
C. H.P.
D. none

Answer

To solve the problem, we start with the given equality:

\[ \frac{a + be^x}{a - be^x} = \frac{b + ce^x}{b - ce^x} = \frac{c + de^x}{c - de^x} \]

Let's denote this common value as \( t \).

### Step 1: Set up the first equation
From the first part of the equality, we have:

\[ \frac{a + be^x}{a - be^x} = t \]

Cross-multiplying gives:

\[ a + be^x = t(a - be^x) \]

### Step 2: Rearranging the equation
Rearranging the equation, we get:

\[ a + be^x = ta - tbe^x \]

Bringing all terms involving \( e^x \) to one side:

\[ be^x + tbe^x = ta - a \]

Factoring out \( e^x \):

\[ e^x(b + tb) = ta - a \]

### Step 3: Isolate \( e^x \)
Now, we can isolate \( e^x \):

\[ e^x = \frac{ta - a}{b(1 + t)} \]

### Step 4: Set up the second equation
Now, we consider the second part of the equality:

\[ \frac{b + ce^x}{b - ce^x} = t \]

Cross-multiplying gives:

\[ b + ce^x = t(b - ce^x) \]

Rearranging:

\[ b + ce^x = tb - tce^x \]

Bringing all terms involving \( e^x \) to one side:

\[ ce^x + tce^x = tb - b \]

Factoring out \( e^x \):

\[ e^x(c + tc) = tb - b \]

### Step 5: Isolate \( e^x \) again
Now, we can isolate \( e^x \):

\[ e^x = \frac{tb - b}{c(1 + t)} \]

### Step 6: Equate the two expressions for \( e^x \)
Since both expressions equal \( e^x \), we can set them equal to each other:

\[ \frac{ta - a}{b(1 + t)} = \frac{tb - b}{c(1 + t)} \]

### Step 7: Simplify the equation
Cancel \( (1 + t) \) from both sides (assuming \( 1 + t \neq 0 \)):

\[ ta - a = \frac{b(tb - b)}{c} \]

### Step 8: Rearranging for ratios
This gives us:

\[ a(t - 1) = \frac{b(tb - b)}{c} \]

### Step 9: Set up the third equation
Now, we consider the third part of the equality:

\[ \frac{c + de^x}{c - de^x} = t \]

Following the same steps as before, we can derive another expression for \( e^x \):

\[ e^x = \frac{tc - c}{d(1 + t)} \]

### Step 10: Equate the expressions for \( e^x \)
Setting this equal to the previous expressions gives us another equation to work with.

### Conclusion
From the derived equations, we can see that the ratios \( \frac{a}{b} \), \( \frac{b}{c} \), and \( \frac{c}{d} \) are all equal, which implies that \( a, b, c, d \) are in geometric progression (GP).

Thus, the final answer is that \( a, b, c, d \) are in GP.